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Topics in Gaussian rough paths theory

vorgelegt vonDiplom-MathematikerSebastian Riedel

Hannover

Von der Fakultat II - Mathematik und Naturwissenschaftender Technischen Universitat Berlin

zur Erlangung des akademischen Grades

Doktor der NaturwissenschaftenDr.rer.nat.

genehmigte Dissertation

Promotionsausschuss:

Vorsitzende: Prof. Dr. Gitta KutyniokBerichter/Gutachter : Prof. Dr. Peter K. FrizBerichter/Gutachter : Prof. Dr. Martin Hairer

Tag der wissenschaftlichen Aussprache: 23. April 2013

Berlin 2013D 83

Berlin, May 5, 2013

Acknowledgement

At first, I would like to express my gratitude to my PhD advisor, Professor Peter Friz, whoconstantly supported me during the time of my doctorate. In particular, I would like to thankPeter for the time he always found for discussing with me and for the patience he had. Hisenduring encouragement laid the basis for the current work.

Next, I would like to thank Professor Martin Hairer for being my second examiner, andProfessor Gitta Kutyniok who kindly agreed to be the chair of the examination board.

I am indebted to all my collaborators who worked with me during the last three years.Namely, I would like to thank Doctor Christian Bayer, Doctor Benjamin Gess, Professor ArchilGulisashvili, Professor Peter Friz, PD Doctor John Schoenmakers and Weijun Xu. Furthermore,I would like to thank Professor Terry Lyons for inviting me to Oxford during my PhD and forthe valuable discussions we had.

This work could have not been written without the financial support of the InternationalResearch Training Group “Stochastic models of complex processes” and the Berlin MathematicalSchool (BMS). I would like to thank all the people working there for their helpfulness andkindness they showed to me during the last years.

Special thanks go to Joscha Diehl, Clement Foucart, Birte Schroder and Maite WilkeBerenguer for reading parts of this thesis and giving valuable comments.

At this point, I would like to mention my colleagues and the friends I met in the mathematicalinstitute of the Technische Universitat Berlin who gave me a very warm welcome and providedan open and friendly atmosphere during the time of my doctorate. In particular, I would liketo thank Professor Michael Scheutzow, my BMS mentor, who gave me a lot of helpful advicesconcerning my PhD. I am also more than thankful to the following people: Michele, who mademe laugh uncountably many times and who introduced me to the dark secrets of pasta andfacebook. To Joscha for the possibility to ask the really relevant questions about rough paths.Thank you, Simon, for many fruitful discussions about and not about math. Maite, thankyou for making me get up on Monday before 7:00 by offering coffee and for the joint exercisesessions. Thanks to Clement for having many beers with me after work and for the first partof Brice de Nice. Last, thank you, Stefano, for B.F.H., the 1st of May and the 2nd chaos.

At the end, I would like to thank my family, in particular my parents, who always had faithin what I am doing. Finally, my biggest thanks go to Birte for encouraging me during the lastyears, for sharing successes and defeats, joy and sorrow, and for chasing math away when itshould not be there.

To Birte

Contents

Introduction 1

Notation and basic definitions 17

1 Convergence rates for the full Brownian rough paths with applications tolimit theorems for stochastic flows 211.1 Rates of Convergence for the full Brownian rough path . . . . . . . . . . . . . . . 24

2 Convergence rates for the full Gaussian rough paths 352.1 Iterated integrals and the shuffle algebra . . . . . . . . . . . . . . . . . . . . . . . 392.2 Multidimensional Young-integration and grid-controls . . . . . . . . . . . . . . . 432.3 The main estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.4 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3 Integrability of (non-)linear rough differential equations and integrals 733.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.2 Cass, Litterer and Lyons revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.3 Transitivity of the tail estimates under locally linear maps . . . . . . . . . . . . . 813.4 Linear RDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.5 Applications in stochastic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4 A simple proof of distance bounds for Gaussian rough paths 914.1 2D variation and Gaussian rough paths . . . . . . . . . . . . . . . . . . . . . . . 934.2 Main estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Spatial rough path lifts of stochastic convolutions 1075.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.2 Conditions in terms of Fourier coefficients . . . . . . . . . . . . . . . . . . . . . . 1135.3 Lifting Ornstein-Uhlenbeck processes in space . . . . . . . . . . . . . . . . . . . . 120

6 From rough path estimates to multilevel Monte Carlo 1316.1 Rough path estimates revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.2 Probabilistic convergence results for RDEs . . . . . . . . . . . . . . . . . . . . . . 1376.3 Giles’ complexity theorem revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.4 Multilevel Monte Carlo for RDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Appendix 151A Kolmogorov theorem for multiplicative functionals . . . . . . . . . . . . . . . . . . . 151

Contents

vi

Introduction

“What I don’t like about measure theory is that you have to say ’almost everywhere’almost everywhere.”

– Kurt Friedrichs

“What is Ω? Cats?”– Michele Salvi

In order to describe correctly the research contributions of this thesis, we begin by a briefhistory of the theory of stochastic integration with a focus on the attempt to define a pathwiseintegral. We will explain the very first goal of the precursors as well as the generalisations madeby some of the leaders in the field. In particular, we will highlight the link between pathwisestochastic integration and Gaussian analysis. The introduction closes with an outline of ourresults.

A basic problem in stochastic calculus is to give a meaning to differential equations of theform

Yt = f(Yt) Xt; Y0 = ξ ∈W, (1)

Y taking values in some Banach space W , X : [0, T ] → V being some random signal withvalues in a Banach space V and f taking values in the space of linear maps from V to W .In a deterministic setting, these equations are also called controlled differential equations. Inmany cases in stochastics, it is natural to assume that X denotes some “noise” term which canformally be written as the differential of a Brownian motion B. However, this causes problemswhen we try to give a rigorous meaning to (1). In fact, a famous property of the trajectoriest 7→ ω(t) of the Brownian motion, i.e. its sample paths, is their non-differentiability on a setof full measure. Therefore, we cannot apply the deterministic theory of controlled differentialequations. One approach is to rewrite the differential equation (1) as an integral equation:

Yt = Y0 +

∫ t

0f(Ys) dXs. (2)

By doing this, we shift the problem of defining (1) to the problem of how to define the (stochas-tic) integral in (2). More generally, we may ask the following question: How can we define astochastic integral of the form

∫ t

0Ys dXs (3)

where X and Y are stochastic processes taking values in V resp. L(V,W )? There are basicallytwo strategies we can follow. The first one ignores all the probabilistic structure the processes Xand Y might have and tries to build up a deterministic theory of integration which is rich enough

Introduction

in order to integrate all sample paths of X and Y with respect to each other. We will call this thepathwise approach. The second strategy uses the probabilistic properties of the processes underconsideration in order to define the integral, and we will call this the probabilistic approach.We will see that Lyons’ rough paths theory can be seen as a pathwise approach, whereas theclassical Ito theory is rather a probabilistic approach.

In the following, we will summarise the most important attempts of defining stochasticintegrals of the form (3) in order to better understand the contribution of rough paths theory inthe context of stochastic integration. This permits us to explain the notion of Gaussian roughpaths which provides the framework for this thesis.

Young’s approach

The first and probably most “natural” pathwise approach is to define the integral (3) as thelimit (at least in probability) of Riemann-sums:∫ t

0Ys dXs = lim

|Π|→0

∑ti∈Π

Yti(Xti+1 −Xti), (4)

where the Π are finite partitions of the interval [0, t]. It is commonly known that this limitexists (pathwise) if the sample paths of X have bounded variation (which is the same as to saythat the sample paths have finite length):

lim|Π|→0

∑ti∈Π

|Xti+1 −Xti | <∞ a.s.

Unfortunately, this is not the case for the Brownian motion and the above quantity will beinfinite almost surely in this case. A more elaborated approach was developed by Laurence C.Young, starting from the article [You36] and further developed in a series of papers. Recall thenotion of p-variation, a generalisation of the concept of bounded variation: If x : [0, T ] → V isa path and p ≥ 1, the p-variation of x is defined as

supΠ

∑ti∈Π

|xti+1 − xti |p 1

p

.

The main theorem of Young can be stated as follows: If x and y are paths of finite p- resp.q-variation with 1

p + 1q > 1, the limit in (4) exists1 and can be bounded in terms of the p- and

q-variation of x and y. Let us note that the condition 1p + 1

q > 1 is necessary; indeed, Younggives a counterexample by constructing paths x and y which have finite 2-variation only andfor which the Riemann sums (4) diverge. Recall now that our initial aim was to solve integralequations of the form

yt = y0 +

∫ t

0f(ys) dxs. (5)

If this equation has a solution, we expect (at least for smooth f) that the solution y has aregularity (on small scales) which is similar to the regularity of x. In other words, if x has finitep-variation, also y and f(y) should have finite p-variation. This means that as long as x hasfinite p-variation for some p < 2, equation (5) should be solvable. That this is indeed the casewas first rigorously worked out - to the author’s knowledge - by Terry Lyons in [Lyo94] for finitedimensional Banach spaces, see also [LCL07] for the general case. Lyons solves equation (5) bya Picard iteration scheme and shows that the solution y varies continuously in x with respect

1Note here that Young considers the case of complex valued paths only; however, the same proof works alsoin Banach spaces, cf. [LCL07].

2

to p-variation topology. Let us go back to stochastics now. If we consider again the Brownianmotion B, it turns out that

supΠ

∑ti∈Π

|Bti+1 −Bti |2 =∞ a.s.

(for a proof, cf. [FV10b, Section 13.9]), hence the sample paths of the Brownian motion donot have finite p-variation (almost surely) for p ≤ 2. In other words, the regularity of thetrajectories slightly fail to fulfill the necessary regularity condition and therefore the theoryof Young integration cannot be applied in the Brownian motion case. This, of course, is aremarkable drawback of Young’s theory. However, it still can be used to solve equations of theform (2) if the trajectories of the driving signal X are “not too rough”; for instance, it applieswhen X is a fractional Brownian motion with Hurst parameter H > 1/2 (the precise definitionof a fractional Brownian motion will be given below).

Ito’s theory of stochastic integration

We will only consider finite dimensional Banach spaces in this paragraph. In the seminalwork [Ito44], Kiyosi Ito was the first who gave a satisfactory definition of the stochastic integral(3) in the case where X is a Brownian motion. In [Ito51], he used this definition to solvedifferential equations driven by a Brownian motion. Since his approach differs very much fromthe pathwise approach, we decided to sketch it briefly. In modern language, the Ito integral isconstructed by first identifying a family of “simple” processes which are piecewise constant, left-continuous and adapted with respect to the Brownian filtration. Adaptedness can be understoodas that at time t, the process does not have more “information” about the Brownian motionthan it provides up to time t (for instance, it cannot look in the future). This is of coursea probabilistic notion. The integral is then defined in a natural way with respect to theseprocesses. One realizes that these simple processes and the stochastic integral both belongto a certain space of processes called martingale spaces, and the stochastic integral defines anisometry between these spaces. Taking the closure in the space of integrable processes thendefines the stochastic integral.

In contrast to the pathwise approach, the Ito integral is now defined as an element in somespace of processes via an isometry. In a second step, one can show that also the Riemann sums(4) converge to this object, but in general only in probability (which is weaker than almost sureconvergence). The theory of Ito integration had an enormous success and is now widely used instochastic calculus. Together with a change of variable formula, called Ito’s Lemma, it providesa powerful tool to solve stochastic differential equations of the form (2) even for more generaldriving processes X. However, the theory has certain constraints. We list two of them:

(i) The class of driving processes is (essentially) limited to (semi-)martingales, i.e. to pro-cesses which have the probabilistic properties of a “fair game”. It is not hard to imaginemodels (e.g. in finance) for which the driving signal does not have this structure.

(ii) Since the integral is defined in a “global” way, it is a priori not clear what happens onthe level of trajectories. Recall that Lyons proved a pathwise continuity for the map x 7→If (x, ξ) := y, y being the solution of (5), when x is a path of finite p-variation for p < 2.However, for Brownian trajectories ω, we do not know which regularity properties themap ω 7→ If (ω, ξ) enjoys. We will actually see that it is not (and cannot be) continuous.

The Ito integral has some unexpected properties. For instance, when replacing the Riemannsums in (4) by ∑

ti∈Π

Yti(Xti+1 −Xti),

3

Introduction

where Yti = (Yti + Yti+1)/2 and then passing to the limit |Π| → 0, we still have convergence(in probability), but not to the same object, which we call the Stratonovich integral in this case.This phenomenon does not occur for the Riemann-Stieltjes or the Young integral. Moreover,the change-of-variable formula (or Ito formula) for the Ito integral contains an additional,unexpected term, the Ito correction term. This term does not occur for the Stratonovichintegral. This already indicates that stochastic integration is very different from the usualintegration theory we know, and that one has some “freedom” when defining the integral.

Follmer’s “Ito formula without probability”

An interesting contribution in the direction of a pathwise approach was made by HansFollmer in the year 1981 in the work [Fol81]. Follmer considers the quadratic variation [x]of a continuous, real valued path x with respect to a sequence of partitions Πn for which themesh-size tends to 0 for n→∞, defined by

limn→∞

∑ti∈Πn; ti<t

|xti+1 − xti |2 =: [x]t

if the right-hand side exists. Follmer shows that for these paths, the Ito-type change of variableformula

f(xt) = f(x0) +

∫ t

0f ′(xs) dxs +

1

2

∫ t

0f ′′(xs) d[x]s

holds for functions f ∈ C2. Note that the definition of the integral on the right hand sidedoes not cause any problems since t 7→ [x]t is increasing, hence has bounded variation and theintegral exists as a Riemann-Stieltjes integral. The formula thus shows the existence of theintegral ∫ t

0f ′(xs) dxs

as the limit of Riemann sums along the sequence of partitions Πn. In the case of the Brownianmotion, it is well-known that for its trajectories ω we have [ω]t = t almost surely for anysequence of nested partitions (Πn)n, hence the integral∫ t

0f ′(Bs) dBs

can be defined in a pathwise manner. The work of Follmer is interesting for us since he founda sufficient criterion, finiteness of the quadratic variation, to define a stochastic integral in apathwise sense.

Lyons’ key insights and the birth of rough paths theory

In the humble opinion of the author, the final breakthrough in the task of defining stochasticintegrals in a pathwise manner was made by Terry Lyons. For a better understanding of theissues, we will first give some “negative” results which show what will not work. Then we willtry to sketch the main ideas of Lyons which constitute what is known today under the term ofrough paths theory.

In the work [Lyo91], Lyons proves the following result: Let C ⊂ C([0, T ],R) be a class ofpaths for which the Ito (or Stratonovich) integral

∫µdν exists (as a limit of Riemann sums)

for all µ, ν ∈ C. Then C has Wiener measure 0. This result shows that even if we managed toextend the definition of the Young integral to a wider class of paths, using, for instance, a finernotion than the notion of p-variation, we would never be able to integrate all Brownian pathswith respect to each other. One could hope that a different and maybe more sophisticateddefinition of the integral might help us getting out of this trouble. That this is not the case is

4

shown by Lyons in [LCL07, Proposition 1.29]: Let B be a Banach space on which the Wienermeasure can be defined in a natural way2. Then there is no bilinear, continuous functionalI : B × B → R for which I(µ, ν) =

∫ 10 µt dνt when µ, ν ∈ B are trigonometric polynomials.

This implies that whatever we choose as a linear3 subspace C ⊂ C([0, T ],R) (where C shouldbe at least rich enough to handle Brownian paths), we will never be able to define a bilinear,continuous functional I : C × C → R (which should be thought of our integral) which fulfills thebasic requirement that I(µ, ν) =

∫ 10 µt dνt holds for all µ, ν ∈ cos(2πn·), sin(2πn·) | n ∈ N.

On the level of controlled differential equations, Lyons proves the following (cf. [LCL07, Section1.5.2]): The map x 7→ If (x, ξ) is not continuous in 2-variation topology. The way Lyons provesthis gives us a first hint what goes wrong for p ≥ 2 and how one might overcome this issue.Lyons defines f in such a way that the solution y = If (x, ξ) is given explicitly as

yt = (y1t , y

2t ) =

(xt − x0,

∫0<u1<u2<t

dxu1 ⊗ dxu2)∈ V ⊕ (V ⊗ V ),

i.e., y consists of the increment of x and its first iterated integral. What he actually proves is thatthe map x 7→

∫0<u1<u2<· dxu1⊗dxu2 is not continuous in 2-variation topology. Loosely speaking,

what we observe here is that a sufficiently rough path does not contain enough information todefine its second (and higher order) iterated integral. However, the above map If (·, ξ) will(trivially) be continuous if we include the information the iterated integral gives us, i.e. if weconsider the map (

x· − x0,

∫0<u1<u2<·

dxu1 ⊗ dxu2)

=: x· 7→ If (x, ξ)

instead. We will see that this kind of continuity indeed holds in much greater generality.The key insight of Lyons is that the path alone does not provide enough information in order

to build up a satisfactory integration theory, but the path together with some extra informationwhich compensates its roughness does. That this extra information is indeed encoded in theiterated integrals (which, we repeat, have to be defined since they are not intrinsically givenin the path) can be seen, for example, from numerical considerations. The solution of a linearcontrolled differential equation can be written down (formally) as a power series:

yt = ξ + f(xt − x0)(ξ) + f⊗2

(∫0<u1<u2<t

dxu1 ⊗ dxu2)

(ξ)

+ . . .+ f⊗n(∫

0<u1<...<un<tdxu1 ⊗ . . .⊗ dxun

)(ξ) + . . . .

For smooth x, this series indeed converges to the solution y. It is quite natural to generalise thissolution concept for non-smooth paths x. In stochastics, the importance of iterated integralsis well-known and widely used for numerical schemes which are known as Taylor schemes (cf.[KP92]).

Lyons first gave a systematic approach to the treatment of rough differential equations in theseminal work [Lyo98]. He begins with the definition of iterated integrals in an abstract setting.Of course, they cannot be just the limit of Riemann sums, but are defined to be objects whichbehave like iterated integrals in an algebraic and analytic way. One could say that these objects“mimic” the iterated integrals of a path x. In his First theorem (or Extension theorem, cf.[Lyo98, Theorem 2.2.1]), he proves that the number of iterated integrals one needs to consideractually depends on the roughness of the path: If x is a path of finite p-variation and its firstbpc iterated integrals are known, then the higher order iterated integrals are uniquely determined.

2cf. [LCL07, Proposition 1.29] for the precise definition here.3We will see that the condition of a linear subspace will be crucial; indeed, our integration theory for rough

paths will not be linear.

5

Introduction

Consequently, he defines a p-rough path x to be the path x together with his first bpc iteratedintegrals. One should note at this point that rough paths spaces are not vector spaces (theyactually cannot be linear if we want to be able to integrate Brownian paths as we saw before)but are metric spaces. The distance of two rough paths x and y takes into account the distancein p-variation between the paths x and y and the higher order iterated integrals. Lyons thendefines a notion of an integral along a rough path x. Note that in general it will not be possibleto integrate two rough paths x and y with respect to each other since the joint integral wouldnecessarily contain mixed integrals of x and y and hence information which is not included inthe respective rough paths. Instead, Lyons first defines the integral over (sufficiently smooth)1-forms α : V → L(V,W ): If x is a p-rough path,∫

α(x) dx,

is defined to be another p-rough path and we have continuity in rough paths topology of themap x 7→

∫α(x) dx. If we can make sense of (x, y) as a joint rough path4, we will be able to

define the integral ∫f(y) dx

for sufficiently smooth functions f : W → L(V,W ) as a rough integral. It turns out that in thesituation of controlled differential equations we can indeed follow this strategy. Lyons solvesthe equation

dyt = f(yt) dxt; y0 = ξ (6)

via a Picard iteration for a p-rough path x and sufficiently smooth f . The solution y is again ap-rough path, and the map x 7→ y =: If (x, ξ) is seen to be continuous in rough paths topology.These results are quite technically involved, but now well understood and outlined in severalmonographs (cf. [LQ98], [LCL07]).

Before we come to the application of rough paths theory in the field of stochastic analysis,we will give some further remarks concerning the deterministic theory.

(i) If (xn) is a sequence of smooth paths, we can solve equation (6) and obtain smoothsolutions (yn). If the iterated integrals of xn converge to a rough path x, continuity of themap x 7→7→ If (x, ξ) implies that also the iterated integrals of yn converge to the solutiony and the limit does not depend on the choice of the initial sequence. This Theorem isknown as the Universal limit theorem. It can be seen as a deterministic analogue of thewell-known Wong-Zakai theorem for Stratonovich stochastic differential equations.

(ii) The statement that the necessary extra information to define a rough path is encodedin its iterated integrals is slightly misleading. In fact, the information is encoded in alliterated integrals indexed by rooted trees (cf. Gubinelli’s work [Gub10] for a clarification).However, the original statement is correct when we define the product of two iteratedintegrals in such a way that the algebra of iterated integrals is isomorphic to the shufflealgebra. In this case, the rough path x has a nice geometric feature; namely, it is seento take values in a Lie group, the free nilpotent group of step bpc over V. Such paths arecalled weakly geometric rough paths. Iterated integrals of smooth paths are also takingvalues in this Lie group and taking the closure with respect to the p-variation metricdefines the space of geometric rough paths. Every geometric rough path is also weaklygeometric, but the converse is false, cf. Friz and Victoir [FV06a]. The geometric pointof view of rough paths theory is worked out in great detail by Friz and Victoir in themonograph [FV10b].

4This is, of course, stronger than just defining x and y as rough paths; the situation can be compared to thefact that the distribution of two random variables X and Y do not determine the joint distribution of (X,Y ).

6

(iii) Although the space of rough paths is not a linear space, one can show that for a fixedreference rough path x, there is a linear space of paths for which all elements can beintegrated with respect to x. These spaces are called spaces of controlled paths and wereintroduced by Gubinelli in [Gub04]. The integration theory for controlled paths is oftenmore flexible and easier to handle than Lyons original integration theory and is now widelyused, see also the forthcoming monograph [FH].

(iv) Rough paths theory was, from the very beginning, closely related to numerical approxima-tion schemes. In the work [Dav07], Davie showed that deterministic Euler- and Milsteinschemes converge to the solution of the respective rough differential equation. This wasgeneralised to step-N Taylor schemes for geometric rough paths by Friz and Victoir in[FV08b], see also [FV10b, Chapter 10].

(v) The map (x, f, ξ) 7→ If (x, ξ), which we will call the Ito-Lyons map in the following, is evenmore regular than we already stated. In fact, it can be seen that it is locally Lipschitzcontinuous in every argument (cf. [FV10b, Chapter 10]). Moreover, the map x 7→ If (x, ξ)is even Frechet differentiable (cf. Li and Lyons [LL06] for the case p < 2 and Friz andVictoir [FV10b, Theorem 11.6] for the general case of geometric rough paths).

Rough paths theory applied to stochastic analysis

Let us go back now to the initial problem of solving controlled differential equations drivenby some random signal X. If we want to apply rough paths theory, we have to say what theiterated integrals of X should be. On the level of trajectories, it is not clear what a “natural”choice of an iterated integral is.5 We will see that taking into account the probabilistic propertiesof the process helps to find a “natural” candidate for an iterated integral.

From now on, we will only consider finite dimensional Banach spaces. In the case of theBrownian motion with independent components, the natural choices for the iterated integralsare the usual Ito and Stratonovich integrals:∫

0<u1<u2<tdBu1 ⊗ dBu2 ,

∫0<u1<u2<t

dBu1 ⊗ dBu2 .

Together with the Brownian process, this indeed defines a process with values in a rough pathsspace, cf. [LQ02], [FV05] or [FV10b, Chapter 13], and we can apply all the machinery providedby rough paths theory. One can also show that the rough path solution one obtains when solving(2) considering the Ito integral coincides with the solution given by Ito’s classical theory; thesame is true for the Stratonovich integral. In a similar fashion, it is possible to find naturallifts for continuous Markov processes (cf. [FV08c], [FV10b, Chapter 16]) and semimartingales([CL05], [FV08a] and [FV10b, Chapter 14]), but we will concentrate on Gaussian processes inthe following. As an example, consider a fractional Brownian motion BH with Hurst parameterH ∈ (0, 1), i.e. a centered Gaussian process with independent components and each componenthas the covariance function

R(s, t) =1

2

(|s|2H + |t|2H − |t− s|2H

).

For H = 1/2, R(s, t) = s ∧ t and we obtain the usual Brownian motion, hence the fractionalBrownian motion can be considered as a natural generalisation. There are many attempts todefine a stochastic calculus with respect to a fractional Brownian motion; cf. [BHOZ08] for anice summary. The process is interesting for us since it is not a semimartingale (or a Markovprocess) for H 6= 1/2, thus Ito’s theory of integration does not apply. The sample paths of BH

5However, it can be seen that every path of finite p-variation can be lifted to a geometric rough path, cf.Lyons and Victoir [LV07]. The problem is that this lift is not (and cannot be) unique.

7

Introduction

are seen to be α-Holder continuous for every α < H, hence we can apply Young’s integrationtheory for H > 1/2. For H ≤ 1/2, the sample paths fail to be α-Holder for α > 1/2. Thequestion now is: are there still “natural” choices of iterated integrals with respect to a fractionalBrownian motion in the case H < 1/2? The first article which gave an answer to this questionis the work of Coutin and Qian [CQ02]. The authors consider the sequence (Πn) of dyadicpartitions of the interval [0, t] and define the process BH(n) to be the process BH for which thesample paths are piecewise linear approximated at the points Πn. Considering the process(

BHt (n)−BH

0 (n),

∫0<u1<u2<t

dBHu1(n)⊗ dBH

u2(n)

),

where the integral exists as a usual Riemann-Stieltjes integral, they show that for H > 1/3,this is a Cauchy sequence almost surely in rough paths topology. The limiting object is definedto be the rough path lift of BH . The authors also consider the third iterated integral andprove convergence, hence there is also a lift for H > 1/4. However, for H = 1/4, they canshow that the second iterated integral diverges. This notion of a rough path lift of a fractionalBrownian motion is indeed quite natural at least for two reasons. First, for H = 1/2, we find theusual Stratonovich integral. Second, it generalises the Wong-Zakai theorem for the fractionalBrownian motion: If Y (n) denotes the solution of the random controlled differential equation

dY (n)t = f(Y (n)t) dBHt (n); Y (n) = Y ∈ ξ,

by the universal limit theorem, Y (n) converges almost surely in rough paths topology to a limitwhich is precisely the solution Y of the corresponding rough differential equation (projectedto the first tensor level). We would like to mention at this point that there are also differentapproaches to define a rough path lift to a fractional Brownian motion (cf. e.g. [Unt09] andthe following articles by the same author), but we will not comment on these approaches here.

Gaussian rough paths in the sense of Friz–Victoir

In [FV10a], Friz and Victoir generalise the method of Coutin and Qian and give a sufficientcriterion on the covariance function R under which a given Gaussian process can be lifted“in a natural way” to a process with sample paths in a rough paths space. In this thesis,we will always work in their framework, therefore we decided to sketch their main ideas here.Let X = (X1, . . . , Xd) be a d-dimensional Gaussian process with independent and identicallydistributed6 components. The main problem is to make sense of the integral∫ t

0(Xi

s −Xi0) dXj

s

for i 6= j. If the trajectories of X are differentiable, we can formally calculate the secondmoment:

E∣∣∣∣∫ t

0(Xi

s −Xi0) dXj

s

∣∣∣∣2 = E∫ t

0

∫ t

0(Xi

s −Xi0)(Xi

u −Xi0)∂sX

js∂uX

ju ds du

=

∫[0,t]2

E[(Xis −Xi

0)(Xiu −Xi

0)] ∂s∂uE[XjsX

ju] ds du

=

∫[0,t]2

R(s, u)−R(s, 0)−R(0, u) +R(0, 0) dR(s, u),

where R denotes the covariance function and the right hand side is a suitable version of a2D Young integral. Fortunately, there is indeed a theory for 2 dimensional Young integration(developed by Towghi in [Tow02]) and we can bound the right hand side in terms of the 2

6The assumption that the components should have the same distribution is not really necessary and onlyassumed for the sake of simplicity.

8

dimensional ρ-variation of R provided ρ < 2. Natural approximations of the sample pathsof the process X (such as a piecewise linear approximation or the convolution with a smoothfunction) yield approximations of the covariance function for which the ρ-variation is seen tobe uniformly bounded. The following result should therefore not come as a surprise: Assumethat the covariance function of every component of X has finite ρ-variation for some ρ < 2.Then there exists a natural lift of X to a process with values in a rough paths space. The liftof the process will be denoted by X in the following. The results of Friz and Victoir are sharpin the sense that the covariance function of a fractional Brownian motion is seen to have finiteρ-variation for ρ = 1

2H , but not better. The threshold ρ = 2 therefore corresponds to the Hurstparameter H = 1

4 for which Coutin and Qian already showed that the natural approximationof the second integral diverges.

Once the existence of a Gaussian rough paths lift is established under this very generalcondition, it can be shown that many theorems from stochastic analysis proven for the Brownianmotion generalise to Gaussian rough paths. For instance, in the article [FV10a], Friz and Victoirprove Fernique estimates for the lift X (see also the work of Friz and Oberhauser [FO10] fora different proof of this result). A support theorem for Gaussian rough paths is proven in[FV10a, Theorem 55]. A large deviation principle for the lift of a fractional Brownian motionwas proven by Millet and Sanz-Sole in [MSS06] and later generalised for Gaussian rough pathsby Friz and Victoir in [FV10b, Theorem 15.55]. A Malliavin-type calculus was established (cf.Cass, Friz and Victoir [CFV09], Friz and Victoir [FV10b, Chapter 20], Cass and Friz [CF11])and a Hormander-type theorem for Gaussian rough paths can be proven (cf. Cass, Friz [CF10],Cass, Litterer, Lyons [CLL], Cass, Hairer, Litterer and Tindel [CHLT12]).

The results of this thesis

We will now summarise the main contributions of this thesis. More details to the respectiveresults may be found in the beginning of the corresponding chapters.

In Chapter 1, we consider the Brownian rough paths lift B, seen as the Brownian incre-ments of a Brownian motion B : [0, 1] → Rd together with its iterated Stratonovich integrals.By Lyons extension theorem, we can lift the sample paths of B to any p-rough paths space pro-vided p > 2. If we approximate the trajectories of the underlying Brownian motion piecewiselinear at the points 0 < 1/n < 2/n < · · · < 1, we obtain another process Bn with piecewiselinear trajectories. This process can be lifted to a process Bn with sample paths in a p-roughpaths space using Riemann-Stieltjes theory. The first result is the following.

Theorem I. For all p > 2 and η < 12 −

1p ,

ρ 1p−Hol(B,Bn) ≤ C

(1

n

)ηalmost surely for all n ∈ N where C is a finite random variable.

ρ 1p−Hol(·, ·) denotes a rough paths metric here. Note that the convergence rate increases for

large p but does not exceed 12 . From the local Lipschitz continuity of the Ito–Lyons map, we

immediately obtain convergence rates for the Wong–Zakai theorem. Moreover, for sufficientlysmooth vector fields, the solution flow of a rough differential equation is differentiable and ourconvergence rates for the Wong–Zakai theorem also hold true on the level of flows:

Theorem II. Let f = (f0, f1, . . . , fd) be smooth vector fields and consider the (random) flowy0 7→ UBn,t←0 (y0) on Re defined by

dy = f0 (y) dt+d∑i=1

fi (y) dBin, y (0) = y0.

9

Introduction

Then a.s.

UBn,t←0 (y0)

converges uniformly (as do all its derivatives in y0) on every compact subset K ⊂ [0,∞)× Rd;and the limit

UB,t←0 (y0) := limn→∞

UBn,t←0 (y0)

solves the Stratonovich SDE

dy = f0 (y) dt+

d∑i=1

fi (y) dBi, y (0) = y0.

Moreover, for every η < 1/2 and every k ∈ 1, 2, . . . and K ⊂ [0,∞)×Rd, there exists an a.s.finite random variable C such that

maxα=(α1,...,αe)

|α|=α1+···+αe≤k

|∂αUB,·←0 (·)− ∂αUBn,·←0 (·)|∞;K ≤ C(

1

n

)η

for all n ∈ N.

Note that this implies an almost sure Wong–Zakai convergence rate of (almost) 12 , which is

known to be sharp (modulo possible logarithmic corrections).The results in this chapter were obtained in collaboration with Prof. Peter Friz and are

published in the journal Bulletin des Sciences Mathematiques, see [FR11].

In Chapter 2, we generalise the results of Chapter 1 to lifts of Gaussian processes X inthe sense of Friz–Victoir. Again, Xn denotes the process with piecewise linear approximatedtrajectories. Our main theorem can be stated as follows:

Theorem III. Assume that the covariance of X has finite ρ-variation in 2D sense and that

the ρ-variation over every square [s, t]2 ⊂ [0, 1]2 can be bounded by a constant times |t − s|1ρ .

Then for all η < 1ρ −

12 and p > 2ρ

1−2ρη ,

ρ 1p−Hol(X,Xn)→ 0

for n→∞ almost surely and in Lq for any q ≥ 1, with rate η.

Note again that a good convergence rate forces p to be chosen large. Note also that ourtheorem holds for much more general approximations than piecewise linear approximations.As a consequence, we obtain almost sure convergence rates for the Wong–Zakai theorem forGaussian rough paths.

Corollary IV. Let f = (f0, f1, . . . , fd) be smooth vector fields and consider the random con-trolled differential equation

dYn = f0(Yn) dt+d∑i=1

fi(Yn) dXin; Yn(0) = ξ.

Then a.s.

Yn → Y

uniformly for n → ∞ with rate η for any η < 1ρ −

12 and the limit solves the random rough

differential equation

dY = f(Y ) dX; Y (0) = ξ.

10

Recall that Davie presented a step-2 Taylor scheme for solving rough differential equations(RDEs) and computed the convergence rate (cf. [Dav07]). Step-N schemes with convergencerates are considered in Friz and Victoir [FV10b, Chapter 10]. In [DNT12], Deya, Neuenkirch andTindel present a simplified Milstein-type scheme for solving rough differential equations drivenby a fractional Brownian motion. The advantage of this numerical scheme is that the iteratedintegrals (which are hard to simulate numerically) are replaced by a product of increments. Ourresults imply sharp convergence rates for these schemes in a general Gaussian setting.

Corollary V. The approximation Yn obtained by running a simplified step-3 Taylor scheme7

with mesh size 1/n for solving the random rough differential equation

dY = f(Y ) dX; Y (0) = ξ

converges almost surely uniformly to the solution Y with rate η for any η < 1ρ −

12 .

This proves a conjecture stated by Deya, Neuenkirch and Tindel in the work [DNT12].

The results in this chapter were obtained in collaboration with Prof. Peter Friz and areaccepted for publication by the journal Annales de l’Institut Henri Poincare Probabilites etStatistiques, see [FR].

In Chapter 3, we consider the work [CLL] of Cass, Litterer and Lyons. Before we describetheir results, it will be useful to make the following definition. Recall that for each Gaussianprocess X, there is an associated Cameron–Martin space (or reproducing kernel Hilbert space).

Definition VI. We say that complementary Young regularity holds for the trajectories of aGaussian process and its Cameron–Martin paths if the Cameron–Martin space is continuouslyembedded in the space of paths which have finite q-variation, the trajectories of X have finitep-variation almost surely and

1

p+

1

q> 1.

The condition assures that we can make sense of the Young integral between the Cameron–Martin paths and the trajectories of the process. In [FV06b, Corollary 1], Friz and Victoir showthat complementary Young regularity holds for the fractional Brownian motion with Hurstparameter H > 1

4 and from their work [FV10a, Proposition 17] it follows that complementaryYoung regularity holds for a Gaussian process X for which the covariance has finite ρ-variationfor ρ < 3

2 .

The aim of the article [CLL] is to prove that the Jacobian of a Gaussian RDE flow hasfinite Lq moments8 for every q ≥ 1. They introduce a map which assigns an integer Nα(x) toa p-rough path x (which equals the number the p-th power of the p-variation of x exceeds thebarrier α). The main work of [CLL] is to show that if we replace the rough paths x by thelift of a Gaussian process X, this number has tails which are strictly “better” than exponentialtails. More precisely, Nα(X) is seen to have Weibull tails with shape parameter strictly greaterthan 1 provided the trajectories of the underlying Gaussian process X and its Cameron–Martinpaths have complementary Young regularity.

Our first contribution is the identification of so-called locally linear maps Ψ, mapping fromone rough paths space to another, under which the tail estimates remain valid. Our result ispurely deterministic.

7In the case ρ = 1, a step-2 scheme converges with the same rate.8The motivation for this is that this result can be used to prove that the solution of a Gaussian rough

differential equation has a smooth density with respect to Lebesque measure at every fixed time point t, cf. also[CHLT12].

11

Introduction

Theorem VII. Let Ψ be a locally linear map. Then there is a α′ such that

Nα′(Ψ(x)) ≤ Nα(x).

Since the p-variation of x can be bounded by a constant times Nα(x), the tail estimatesobtained for Nα(x) remain valid for the p-variation of x. Rough integration and the Ito-Lyonsmap are examples of locally linear maps; hence we immediately obtain

Corollary VIII. Assume that complementary Young regularity holds for the trajectories of Xand its Cameron–Martin paths. Then the following objects have exponential tails9:

(i) The rough integral ∫α(X) dX

where α is a suitable one-form.

(ii) The p-variation of Y where Y solves the random rough differential equation

dY = f(Y) dX; Y (0) = ξ

for smooth and bounded vector fields f .

For linear vector fields f ∈ L(W,L(V,W )) ∼= L(V,L(W,W )), the situation is different.

Theorem IX. If y solves the linear rough differential equation

dy = f(y) dx = f(dx)(y); y(0) = ξ, (7)

then

Nα(y) ≤ C(1 + ξ)p exp(CNα(x))

for a constant C. In particular, if Nα(X) has Weibull tails with shape parameter strictly greaterthan 1 (which is the case if complementary Young regularity holds for the trajectories of Xand its Cameron–Martin paths), the p-variation of the solution Y of the random linear roughdifferential equation (7) has finite Lq momemts for any q ≥ 1.

Our estimates particularly imply that the Jacobian of a Gaussian RDE flow has finite Lq

moments for any q ≥ 1, which was the main result of the work [CLL]. The estimates are alsorobust in the sense that they can be used to prove uniform tail estimates. As an example, weshow that a certain rough integral over a family of Gaussian processes has uniformly Gaussiantails, a technical result needed by Hairer in [Hai11].

The results in this chapter were obtained in collaboration with Prof. Peter Friz and arepublished in the journal Stochastic Analysis and Applications, see [FR13].

In Chapter 4, we apply the methods from Lyons and Xu presented in [LX12] to boundthe distance between two Gaussian rough paths in p-variation topology. Our estimates are verysimilar to the ones needed for proving Theorem III, but we show how to avoid the algebraicmachinery presented in Chapter 2 and still get optimal bounds in the case ρ = 1. Our maintheorem states the following:

9In fact, our tail estimates are sharper and can be expressed in terms of Weibull tails, cf. Chapter 3. Werestrict ourselves to exponential tails for the sake of simplicity.

12

Theorem X. Let (X,Y ) = (X1, Y 1, · · · , Xd, Y d) be a jointly Gaussian process and let (Xi, Y i)and (Xj , Y j) be independent for i 6= j. Assume that there is a ρ ∈ [1, 3

2) such that the ρ-variation of R(Xi,Y i) is bounded by a constant K for every i = 1, . . . , d. Let γ ≥ ρ such that1γ + 1

ρ > 1. Then, for every p > 2γ, q ≥ 1 and δ > 0 small enough, there exists a constant CKsuch that

(i) if 12γ + 1

ρ > 1, then

|ρp−var(X,Y)|Lq ≤ C supt|Xt − Yt|

1− ργ

L2 , (8)

(ii) if 12γ + 1

ρ ≤ 1, then

|ρp−var(X,Y)|Lq ≤ C supt|Xt − Yt|3−2ρ−δ

L2 . (9)

In our theorem, ρp−var(·, ·) denotes a p-rough path metric. Note that for ρ = 1, we canalways use the estimate (8). Inequality (8) is actually valid for all γ ≥ ρ provided 1

γ + 1ρ > 1

which can be seen by using the techniques developed in Chapter 2, but one aim of Chapter4 is to show that we can avoid a bulk of calculations and still obtain estimate (9) (which isnot sharp though). One can show that our results imply convergence rates as in Theorem III,and for ρ = 1 we obtain optimal convergence. Another application of Theorem X appears inthe field of stochastic partial differential equations. In [Hai11], Hairer considers the stationarysolution ψ of the equation

dψ = (∂xx − 1)ψ dt+ σ dW (10)

where σ is a positive constant, the spatial variable x takes values in [0, 2π], ∂xx is equippedwith periodic boundary conditions and dW is space-time white noise, i.e. a standard cylindricalWiener process on L2([0, 2π],R). Hairer shows that for every fixed time point t, the Gaussianprocess ψt obtained by taking d independent copies of the spatial processes ψt can be lifted toa process Ψt with values in a p-rough paths space for any p > 2. He also shows that there is acontinuous modification of the map t 7→ Ψt. Our results imply optimal time regularity.

Corollary XI. There is an α-Holder continuous modification of the map t 7→ Ψt for everyα < 1

4 −12p .

Note that the Holder exponent increases for large p and is bounded by 1/4 which is knownto be a sharp bound.

The results in this Chapter were obtained in collaboration with Weijun Xu and are availableonline, see [RX12].

In Chapter 5, we reinvestigate the solution of the modified heat equation (10). The spaceregularity of ψt essentially depends on two factors: the smoothing effect of the operator ∂xxand the “colouring” of the noise dW . We have already seen that the crucial condition forlifting ψt to a process in a rough paths space (in the sense of Friz–Victoir) is a sufficientlyregular covariance function Rψt in terms of 2 dimensional ρ-variation. The parameter ρ shouldtherefore also depend on the smoothing effect of the operator and the colouring of the noise.Our main theorem determines the parameter ρ for which the ρ-variation of Rψt is finite in termsof the spectrum of the operator and the noise. For simplicity, we only state the result for thefractional heat equation here.

13

Introduction

Theorem XII. Let ψ be the stationary solution of the fractional, modified heat equation

dψ = (−(−∂xx)α − 1)ψ dt+ σ dW ; α ∈ (1/2, 1] (11)

with periodic boundary conditions where dW is space-time white noise. Then the ρ-variationof Rψ is finite for ρ = 1

2α−1 . In particular, if we set ψα := (ψ1, . . . , ψd) where the ψi are

independent copies of ψ, for every fixed t we can lift the trajectories of ψαt to p-rough pathsin the sense of Friz–Victoir for all p > 2

2α−1 provided α > 3/4. Moreover, there is a Holdercontinuous modification of the lifted process t 7→ Ψα

t .

In addition, our results imply uniform bounds of the ρ-variation for viscosity and Galerkinapproximations of (11) which can be used in a future work for numerical considerations. Wealso give a new and easy criterion on the covariance of a Gaussian process with stationaryincrements to have finite ρ-variation. If the process is given as a random Fourier series (as inthe situation above), these conditions translate into conditions on the Fourier coefficients.

The results in this chapter were obtained in collaboration with Prof. Peter Friz, Dr. Ben-jamin Gess and Prof. Archil Gulisashvili and are available online, see [FGGR12].

In Chapter 6 we come back to numerical considerations. Let Y be the solution of a randomrough differential equation

dY = f(Y ) dX; Y (0) = ξ, (12)

X being the lift of a Gaussian process whose covariance is of finite ρ-variation. In Corollary V,we saw that there is an easy implementable numerical scheme which converges almost surely tothe solution of (12). Let Yn denote an approximation of Y using such a scheme with mesh-size1/n. Assume now that we are interested in evaluating a quantity of the form Eg(Y ) where g isa functional which may depend on the whole path of Y . The first obstacle is that we do notknow, even for smooth g, with which rate Eg(Yn) converges to Eg(Y ) for n→∞ since we onlyproved almost sure convergence, not L1 convergence (which would imply a convergence ratewhen g is at least Lipschitz). This is our first result.

Theorem XIII. Assume that complementary Young regularity holds for the trajectories ofX and its Cameron–Martin paths. Then the Wong-Zakai approximation in Corollary IV andthe simple Taylor scheme in Corollary V both converge in Lq for any q ≥ 1 with the sameconvergence rate.

We would like to mention here that the proof of this theorem is more involved than onemight expect at a first sight. In fact, we improve the estimate for the Lipschitz constant ofthe Ito–Lyons map slightly, using similar estimates as in Chapter 3 for the case of linear roughdifferential equations, and then use the results of Cass, Litterer and Lyons in [CLL] to provethe assertions. As an immediate corollary of Theorem XIII, we obtain strong convergence ratesfor our numerical scheme in the case when g is Lipschitz. However, at the present stage, we canonly bound the weak convergence rate from below with the strong rate, whereas the weak ratemight be better, at least for smooth g. If we want to evaluate Eg(Y ), a Monte–Carlo evaluationwould be a possible and easy method. In the seminal work [Gil08b], Giles showed that one canreduce the computational complexity (more precisely: its asymptotics for a given mean squarederror) dramatically when using a multilevel Monte Carlo method. For us, the multilevel methodis also interesting because the strong convergence rate plays a more important role here than theweak one (which would be used calculating the complexity of the usual Monte Carlo evaluation).Indeed, we can prove an abstract, more general complexity theorem as in [Gil08b] which fitsour purposes. Applied to the evaluation of Eg(Y ), we can prove the following result.

Theorem XIV. Assume that complementary Young regularity holds for the trajectories of Xand its Cameron–Martin paths and that g is Lipschitz. Then the Monte Carlo evaluation of

14

a path-dependent functional of the form Eg(Y ), to within a mean squared error of ε2, can beachieved with computational complexity

O(ε−θ)∀θ > 2ρ

2− ρ.

In the case of a Brownian motion, the asymptotics of the computational complexity isbounded by O

(ε−θ)

for any θ > 2 which is known to be sharp modulo logarithmic corrections,cf. [Gil08a, Gil08b]. Compared to a usual Monte Carlo method, we see that indeed a multilevelmethod decreases the computational complexity in the general Gaussian setting.

The results in this chapter were obtained in collaboration with Dr. Christian Bayer, Prof. Pe-ter Friz and PD Dr. John Schoenmakers.

15

Introduction

16

Notation and basic definitions

In this chapter, we introduce the most important concepts and definitions from rough paththeory. For a detailed account, we refer to [FV10b], [LCL07] and [LQ02].

Fix a time interval [0, T ]. For all s < t ∈ [0, T ], we define the n-simplex

∆ns,t := (u1, . . . , un) | s < u1 . . . < un < t.

We will simply write ∆s,t instead of ∆2s,t and ∆ instead of ∆0,T . Let (E, d) be a metric space

and x ∈ C([0, T ], E). For p ≥ 1 and α ∈ (0, 1] we define

‖x‖p−var;[s,t] := supD⊂[s,t]

∑ti,ti+1∈D

|d(xti , xti+1)|p 1

p

and ‖x‖α−Hol;[s,t] := sup(u,v)∈∆s,t

d (xu, xv)

|v − u|α

where D ⊂ [s, t] means that D is a finite dissection of the form s = t0 < . . . < tM = t of theinterval [s, t]. We will use the short hand notation ‖·‖p−var and ‖·‖α−Hol for ‖·‖p−var;[0,T ] resp.‖·‖α−Hol;[0,T ] which are easily seen to be semi-norms.

Given a positive integer N , the truncated tensor algebra of degree N is given by the directsum

TN (Rd) = R⊕ Rd ⊕(Rd ⊗ Rd

)⊕ ...⊕

(Rd)⊗N

=N⊕n=0

(Rd)⊗n

and we will write πn : TN(Rd)→(Rd)⊗n

for the projection on the n-th tensor level. TN(Rd)

isa (finite-dimensional) R-vector space. For elements g, h ∈ TN

(Rd), we define g ⊗ h ∈ TN

(Rd)

by

πn (g ⊗ h) =

n∑i=0

πn−i (g)⊗ πi (h) .

One can easily check that(TN

(Rd),+,⊗

)is an associative algebra with unit element e :=

1 + 0 + 0 + . . .+ 0 . We call it the truncated tensor algebra of level N . A norm is defined by

|g|TN(Rd) = maxn=0,...,N

|πn (g)|

which turns TN(Rd)

into a Banach space.A continuous map x : ∆ → TN

(Rd)

is called multiplicative functional if for all s < u < tone has xs,t = xs,u ⊗ xu,t. For a path x =

(x1, . . . , xd

): [0, T ]→ Rd and s < t, we will use the

notation xs,t = xt − xs. If x has bounded variation (or finite 1-variation), we define its n-thiterated integral by

xns,t =

∫∆ns,t

dx⊗ . . .⊗ dx

=∑

1≤i1,...,in≤d

∫∆ns,t

dxi1 . . . dxinei1 ⊗ . . .⊗ ein ∈(Rd)⊗n

Notation, basic definitions

where e1, . . . , ed denotes the Euclidean basis in Rd and (s, t) ∈ ∆. The canonical liftSN (x) : ∆→ TN

(Rd)

is defined by

πn

(SN (x)s,t

)=

xns,t if n ∈ 1, . . . , N1 if n = 0.

It is well known (as a consequence of Chen’s theorem) that SN (x) is a multiplicative functional.

Set xt := SN (x)0,t. One can show that xt really takes values in

GN(Rd)

=g ∈ TN

(Rd)

: ∃x ∈ C1-var(

[0, 1] ,Rd)

: g = SN (x)0,1

,

a submanifold of TN(Rd), called the free step-N nilpotent Lie group with d generators. The

dilation operator δ : R×GN(Rd)→ GN

(Rd)

is defined by

πi (δλ(g)) = λiπi(g), i = 0, ..., N.

The Carnot-Caratheodory norm, given by

‖g‖ = inf

length(x) : x ∈ C1-var(

[0, 1] ,Rd), SN (x)0,1 = g

defines a continuous norm on GN

(Rd), homogeneous with respect to δ. This norm induces a

(left-invariant) metric on GN(Rd)

known as Carnot-Caratheodory metric,

d(g, h) :=∥∥g−1 ⊗ h

∥∥ .Let x,y ∈ C0

([0, T ], GN

(Rd))

, the space of continuous GN(Rd)-valued paths started at

the neutral element. We will use the canonical notion of increments expressed by

xs,t := x−1s ⊗ xt.

We define p-variation- and α-Holder-distances by

dp−var;[s,t](x,y) := supD⊂[s,t]

∑ti,ti+1∈D

|d(xti,ti+1 ,yti,ti+1)|p 1

p

and

dα−Hol;[s,t](x,y) := sup(u,v)∈∆s,t

d (xu,v,yu,v)

|v − u|α.

As usual, we set dp−var := dp−var;[0,T ] and dα−Hol := dα−Hol;[0,T ]. Note that dα−Hol(x, 0) =‖x‖α−Hol and dp−var(x, 0) = ‖x‖p−var where 0 denotes the constant path equal to the neutralelement. These metrics are called homogeneous rough paths metrics. We define the followingpath spaces:

(i) Cp−var0

([0, T ] , GN

(Rd))

: the set of continuous functions x from [0, T ] into GN(Rd)

suchthat ‖x‖p−var <∞ and x0 = e.

(ii) Cα−Hol0

([0, T ] , GN

(Rd))

: the set of continuous functions x from [0, T ] into GN(Rd)

suchthat ‖x‖α−Hol <∞ and x0 = e.

(iii) C0,p−var0

([0, T ] , GN

(Rd))

: the dp−var-closure ofSN (x) , x : [0, T ]→ Rd smooth

.

18

(iv) C0,α−Hol0

([0, T ] , GN

(Rd))

: the dα−Hol-closure ofSN (x) , x : [0, T ]→ Rd smooth

.

If N = bpc, the elements of the spaces (i) and (ii) are called weak geometric (Holder) roughpaths, the elements of (iii) and (iv) are called geometric (Holder) rough paths. It is clear bydefinition that every p-rough path is also a multiplicative functional.

By Lyon’s First Theorem (or Extension Theorem, see [Lyo98, Theorem 2.2.1] or [FV10b,Theorem 9.5]) every p-rough path (or 1

p -Holder rough path) x has a unique lift to a path with

finite p-variation in GN(Rd)

for any N ≥ [p]. We denote this lift by SN (x) and call it theLyons lift. Note that the map SN is continuous in rough paths topology. For p-rough pathsand 1

p -Holder rough paths x, we will also use the notation

xns,t :=

∫∆ns,t

dx⊗ · · · ⊗ dx := πn

(SN (x)s,t

)for N ≥ n. This is consistent with our former definition in the case where x had boundedvariation. An immediate consequence of the extension theorem is that any p-rough path x canbe lifted to a q-rough path for every q ≥ p. We will sometimes abuse notation and use the sameletter x to denote this lift.

In this work, we will often be interested in inhomogeneous rough paths metrics which weaim to define now. First, recall that a control is a function ω : ∆ → R+ which is continuousand super-additive in the sense that for all s ≤ u ≤ t one has

ω(s, u) + ω(u, t) ≤ ω(s, t).

If ω is a control, we define

‖x‖p−ω;[s,t] := sups≤u<v≤t

‖xu,v‖ω(u, v)1/p

(with the convention 0/0 := 0)

ρ(k)p−ω;[s,t](x,y) := sup

s≤u<v≤t

|πk(xu,v − yu,v)|ω(u, v)k/p

ρNp−ω;[s,t](x,y) := maxk=1,...,N

ρ(k)p−ω;[s,t](x,y)

ρ(k)p−var;[s,t](x,y) := sup

(ti)⊂[s,t]

(∑i

∣∣πk (xti,ti+1 − yti,ti+1

)∣∣p/k)k/pρNp−var;[s,t](x,y) := max

k=1,...,Nρ

(k)p−var;[s,t](x,y).

where N ≥ bpc. If N = bpc, ρNp−ω and ρNp−var define rough paths metrics and we will write ρp−ωand ρp−var in that case. Note that the metrics dp−var and ρp−var both induce the same topology

on the rough paths spaces, as do the metrics d 1p−Hol and ρp−ω on C

1p−Hol

([0, T ], GN (Rd)) with

the choice ω(s, t) = |t− s|; cf. [FV10b] for more details.We will now define two notions of two-dimensional p-variation. Let A = [a, b]×[c, d] ⊆ [0, T ]2

be a rectangle. If a = b or c = d we call A degenerate. Two rectangles are called essentiallydisjoint if their intersection is empty or degenerate. A partition Π of a rectangle R ⊆ [0, T ]2 isa finite set of essentially disjoint rectangles, whose union is R; the family of all such partitionsis denoted by P (R). A rectangular increment of a function f : [0, T ]2 → R is defined in termsof f evaluated at the four corner points of A,

f (A) := f

(a, bc, d

):= f(b, d)− f(b, c)− f(a, d) + f(a, c).

19

Notation, basic definitions

Recall that a dissection D of an interval [a, b] ⊂ [0, T ] is of the form

D = a = t0 < t1 < · · · < tn = b

and we will write D ([a, b]) for the family of all such dissections. For a function f : [0, T ]2 → Rand p ≥ 1 we set

Vp (f ; [s, t]× [u, v]) := supD=(ti)∈D([s,t])

D′=(t′j)∈D([u,v])

∑i,j

∣∣∣∣f ( ti, ti+1

t′j , t′j+1

)∣∣∣∣p 1

p

,

|f |p-var;[s,t]×[u,v] := supΠ∈P([s,t]×[u,v])

(∑A∈Π

|f (A)|p)1/p

.

We say that f has finite p-variation if Vp(f ; [0, T ]2) < ∞ and finite controlled p-variation if|f |p-var;[0,T ]2 < ∞. If in addition |f |p

p-var;[s,t]2≤ C|t − s| for a constant C and all [s, t] ⊆ [0, T ]

we say that f has finite Holder controlled p-variation. The difference of 2D p-variation andcontrolled p-variation is that in the former, one only takes the supremum over grid-like partitionswhereas in the latter, one takes the supremum over all partitions of the rectangle.

The reason for introducing these two notions of 2D p-variation comes from the fact that[s, t] × [u, v] 7→ Vp(f ; [s, t] × [u, v])p will in general not be superadditive, hence does not definea 2D control function: A function ω : ∆×∆→ R+ is called a (2D) control if it is continuous,zero on degenerate rectangles and super-additive in the sense that for all rectangles R ⊂ [0, T ]2,

n∑i=1

ω (Ri) ≤ ω (R)

whenever Ri : i = 1, . . . , n ∈ P (R). ω is called symmetric if ω ([s, t]× [u, v]) = ω ([u, v]× [s, t])holds for all s < t and u < v. We say that the p-variation of f is controlled by ω if|f (R)|p ≤ ω (R) holds for all rectangles R ⊂ [0, T ]2. It is easy to see that if ω is a 2Dcontrol, (s, t) 7→ ω([s, t]2) defines a 1D-control. By superadditivity, the existence of a con-trol ω which controls the p-variation of f implies that f has finite controlled p-variation and|f |p−var;R ≤ ω (R)1/p. In this case, we can always assume w.l.o.g. that ω is symmetric, otherwisewe just substitute ω by its symmetrization ωsym given by

ωsym ([s, t]× [u, v]) = ω ([s, t]× [u, v]) + ω ([u, v]× [s, t]) .

The connection between finite variation and finite controlled p-variation is summarized in thefollowing theorem.

Theorem 0.0.1. Let f : [0, T ]2 → R be continuous and R ⊂ [0, T ]2 be a rectangle.

(i) We haveV1 (f,R) = |f |1−var;R .

(ii) For any p ≥ 1 and ε > 0 there is a constant C = C (p, ε) such that

1

C|f |(p+ε)−var;R ≤ Vp−var (f,R) ≤ |f |p−var;R .

(iii) If f has finite controlled p-variation, then

R 7→ |f |pp−var;Ris a 2D-control. In particular, there exists a 2D-control ω such that for all rectanglesR ⊂ [0, T ]2 we have |f (R)|p ≤ ω (R), i.e. ω controls the p-variation of f .

Proof. [FV11, Theorem 1].

For more details on higher dimensional p-variations we refer to [FV11, FV10b].

20

1

Convergence rates for the fullBrownian rough paths withapplications to limit theorems forstochastic flows

The purpose of this chapter is to give a proof of a quantitative version of a well-known limittheorem for stochastic flows which goes back to Bismut, Kunita, Malliavin, ... (see [Mal97, Thm6.1] and the references therein). It says, in essence, that if one uses piecewise linear approxi-mations to multidimensional Brownian driving signals, the resulting solutions to the (random)ODEs will converge as stochastic flows to the solution of the (Stratonovich) stochastic differen-tial equations; that is, the solution flows and all their derivatives will convergene uniformly oncompacts.

It has been been understood in recent years that rough path theory [Lyo98, LQ02, FV10b] isideally suited to prove such limit theorems; also on the level of flows [LQ98, FV10b]. In fact, in[Mal97, p. 216] Malliavin himself remarked ”Lyons’s forthcoming theory will reduce the proof of[Mal97, Thm 6.1] to a limit theorem for Brownian motion and Levy’s area”. The price one hasto pay for this reduction is that one has to work with refined Holder (or p-variation) metrics onrough path spaces; to wit, if B denotes the Brownian rough paths; i.e. d-dimensional Brownianmotion B and so (d)-valued Levy’s area A written as

B = exp (B +A) = 1 +B +

∫ ·0B ⊗ dB ∈ R⊕ Rd ⊕

(Rd)⊗2

its rough path regularity is summarized in that there exists a 1/α ∈ [2, 3), in fact we may takeany α ∈ (1/3, 1/2), such that1

ρα-Hol;[0,T ] (B,1) ≡ sup0≤s<t≤T

|Bt −Bs||t− s|α

∨ sup0≤s<t≤T

∣∣∣∫ ts (Br −Bs)⊗ dB∣∣∣

|t− s|2α

is almost surely finite. The Ito(-Lyons)-map is known to be continuous (in fact, locally Lipchitzcontinuous) in this rough path metric, also on the level of flows of (e.g. [FV10b, Ch. 11]), hencethe problem is reduced to show that

ρα-Hol;[0,T ] (B,S2 (Bn))→ 0 a.s.

1We set 1 = exp (0) = 1 + 0 + 0 in R⊕ Rd ⊕(Rd)⊗2

.

Rates Brownian rough paths

where Bn = BDn denotes the piecewise linear approximations to B, based on the dyadicsDn = iT/2n : i = 0, 1, . . . , 2n,

S2 (Bn) = 1 +Bn +

∫ ·0Bnr ⊗ Bn

r dr

is the canonical lift of the piecewise smooth sample paths of Bn. There are various ways ofproving this, the most elegant perhaps being the argument of [Mal97, Thm 5.2], combined withinterpolation and a uniformity consideration based on Doob’s maximal inequality [FV10b, Ch.12]; such martingale arguments are not applicable when working with more general piecewiselinear approximations, such as those based on Dn = iT/n : i = 0, 1, . . . , n. Nonetheless adirect computation [HN09] shows that

∀η < 1

2− α : ρα-Hol;[0,T ] (B,S2 (Bn)) = O

((1

n

)η)a.s.

and one can see that this is the best result of this type. As a corollary, local Lipschitzness ofthe Ito(-Lyons) map implies convergence of the corresponding stochastics flows with the samerate η. It is interesting to compare this with known convergence rates of such approximationsas established in the works of Gyongy [GS06] and others; there, it seems to be folklore ofthe subject that convergence takes place with rate η < 1/2. (Similar to the rate of strongconvergence of the Euler-scheme, which is also of rate η < 1/2). At first glance, these resultsare recovered by taking α ↓ 0. Unfortunately, this is not possible here. The problem is that incurrent setting based on N = 2 iterated integrals ρα-Hol;[0,T ] ceases to be a rough path metricfor α ≤ 1/3. Indeed, rough path theory dictates a sharp relationship between the number ofrequired levels, N , and the (α-Holder type) regularity of the signals under consideration:

N = [1/α] ;

see [FO09] for some subtle (counter-)examples in this context. Back to the case N = 2, we areforced into the regime α ∈ (1/3, 1/2]. In particular, the ”best” rate η < 1

2 −α can only be takenarbitrarily close to

1

2− 1

3=

1

6

which leaves a significant gap to the known rate ”anything less than 1/2”. This gap was alsonoted, in the context of fractional Brownian motion, in [DNT12]. Having explained the problemfrom this point of view, the idea to work with general level N , instead of level 2, is not far andthis is precisely what we shall do. To wit, we shall establish

Theorem 1.0.1 (Rates for the full Brownian rough path). Let Bn denote piecewise linear ap-proximation to d-dimensional Brownian motion B based on the dissection Dn = iT/n : 0 ≤ i ≤ nof [0, T ]. For any integer N , α ∈ [0, 1/2) and η < 1

2 − α there exists an a.s. finite randomvariable C (ω) such that

ρα-Hol;[0,T ] (SN (B) ,SN (Bn)) ≤ C (ω)

(1

n

)η.

Proof. By scaling it suffices to discuss T = 1. This is the content of the next section.

The above rates now lead to the following quantitative version of the limit theorem forstochastic flows [Mal97, Thm 6.1].

22

Corollary 1.0.2. Consider, on Re, (d+ 1) C∞-bounded vector fields V0, V1, . . . , Vd. Considerthe (random) flow y0 7→ UBn,t←0 (y0) on Re defined by

dy = V0 (y) dt+d∑i=1

Vi (y) dBin, y (0) = y0

where Bn is the piecewise linear approximation to Brownian motion based on the dissectionDn = iT/n : 0 ≤ i ≤ n of [0, T ]. Then a.s.

UBn,t←0 (y0)

converges uniformly (as do all its derivatives in y0) on every compact subset K ⊂ [0, T ] × Rd;and the limit

UB,t←0 (y0) := limn→∞

UBn,t←0 (y0)

solves the Stratonovich SDE

dy = V0 (y) dt+

d∑i=1

Vi (y) dBi, y (0) = y0.

Moreover, for every η < 1/2 and every k ∈ 1, 2, . . . and K ⊂ [0, T ]× Rd, there exists an a.s.finite random variable C (ω) such that

maxα=(α1,...,αe)

|α|=α1+···+αe≤k

|∂αUB,·←0 (·)− ∂αUBn,·←0 (·)|∞;K ≤ C (ω)

(1

n

)η.

Proof. We recall that UB,t←0 (y0) can be obtained as solution to the rough differential equation(RDE)

dy = V0 (y) dt+

d∑i=1

Vi (y) dB ≡V (y) d (t,B) .

As is well-known, the map B 7→ y is (locally) Lipschitz continuous when regarding B as geo-metric α-Holder rough path, α ∈ (1/3, 1/2), and y a Re-valued α-Holder path. (The presence ofa drift vector field V0 does not affect this; it suffices to remark that the so-called Young-pairingmap

B 7→ (t,B) ,

has similar Lipschitz regularity in rough path metric.) By basic consistency properties of RDEsolutions, y is also the solution of the same RDE driven by the Lyons lift SN (B), any N ≥ 2,and the map SN (B) 7→ y is (locally) Lipschitz continuous when regarding B as geometric α-Holder rough path, α ∈ (1/ (N + 1) , 1/N), and y a Re-valued α-Holder path. Moreover, this(local) Lipschitz regularity persists when one regards the ensemble

∂aUB,·←0 (·) : |a| ≤ k

(here a =(a1, . . . , ae

)denotes a multi-index of order |a| = a1 + · · · + ae) which corresponds

to a (non-explosive! cf. Thm 11.12 in [FV10b]) system of rough differential equations; afterlocalization the entire ensemble solves a (high-dimensional) rough differential equation and weargue as above. Finally, given η < 1/2 we pick α > 0 such that η < 1/2−α and then N := [1/α].From our ”Rates for the full Brownian rough path” and (local) Lipschitz continuity of thesolution map, on the level of stochastic flows, the result follows.

23


1.1 Rates of Convergence for the full Brownian rough path

Note that in the forthcoming section, we will use generic constants c which may depend on thedimension d without explicitly mentioning it. As seen before, we can restrict ourselfes to thecase [0, T ] = [0, 1].

Lemma 1.1.1. Let x be a multiplicative functional in TN(Rd), (s, t) ∈ ∆ and s = t0 < . . . <

tm = t, m ≥ 1. Then

πn(xs,t) =m−1∑i=0

πn(xti,,ti+1) (1.1)

+

m−1∑i=1

n−1∑l=1

πn−l(xs,ti)⊗ πl(xti,ti+1)

for every n = 1, . . . , N .

Proof. Easy, for instance by induction over m.

The next lemma gives an L2-estimate for the higher order iterated Stratonovich integrals ofB.

Lemma 1.1.2. Let B be a Brownian motion in Rd and take n ∈ N. Then there is a constantc depending only on n such that∣∣∣∣∣

∫∆ns,t

dB ⊗ . . .⊗ dB

∣∣∣∣∣L2

≤ c |t− s|n2

for any (s, t) ∈ ∆.

Proof. Follows from Brownian scaling.

LetD = 0 = t0 < t1 < . . . < tm = 1 be a partition of the unit interval. We use the notation|D| = maxi=1,...,m |ti − ti+1| for the mesh-size of D. BD : [0, 1]→ Rd denotes the piecewise linearapproximation of B w.r.t. D, i.e. BD

ti = Bti for ti ∈ D and, for t ∈ [ti, ti+1], BDt is defined via

BDt −BD

ti

t− ti=Bti+1 −Btiti+1 − ti

.

It is clear that BD is a Gaussian process with paths of finite variation. Therefore we can defineits n-th iterated integral ∫

∆n0,1

dBD ⊗ . . .⊗ dBD ∈(Rd)⊗n

in Riemann-Stieltjes sense. We will later show that Lemma 1.1.2 also holds for BD where theconstant c does not depend on the choice of D. Note that for t ∈ (ti, ti+1),

BDt =

Bti+1 −Btiti+1 − ti

D=

B1

(ti+1 − ti)1/2.

Therefore, in the special case (s, t) = (ti, ti+1), we already see that∣∣∣∣∣∫

∆nti,ti+1

dBD ⊗ . . .⊗ dBD

∣∣∣∣∣L2

=

∣∣B⊗n1

∣∣L2

|ti+1 − ti|n2

∫∆nti,ti+1

du1 · · · dun (1.2)

=

∣∣B⊗n1

∣∣L2 |ti+1 − ti|n

|ti+1 − ti|n2 n!

= c |ti+1 − ti|n2

24

Rates of Convergence for the full Brownian rough path

with c =|B⊗n1 |L2

n! which only depends on n.

Next, we proof a technical lemma which we will need for the proof of Lemma 1.1.4. Recallthe definition of the Levy area of a Brownian motion B =

(B1, . . . , Bd

): For (s, t) ∈ ∆,

As,t ∈ Rd ⊗ Rd is defined by

As,t =∑

1≤i,j≤dAi,js,tei ⊗ ej where

Ai,js,t =1

2

(∫ t

sBis,r dB

jr −Bj

s,r dBir

).

Lemma 1.1.3. Let B be a Brownian motion in Rd on a probability space (Ω,F , (Ft) ,P). LetD = s = t0 < . . . < tm = t be a partition of the interval [s, t] ⊂ [0, 1]. Assume that X is a

stochastic process in(Rd)⊗n

and that Xti is Fti-measurable for all ti ∈ D.

(i) There is a constant c1 = c1 (n) such that

∣∣∣∣∣m−1∑i=0

Xti ⊗Bti,ti+1

∣∣∣∣∣L2

≤ c1 maxi=0,...,m−1

|Xti |L2 |t− s|1/2 .

(ii) There is a constant c2 = c2 (n) such that

∣∣∣∣∣m−1∑i=0

Xti ⊗Ati,ti+1

∣∣∣∣∣L2

≤ c2 |D|1/2 maxi=0,...,m−1

|Xti |L2 |t− s|1/2 .

Proof. (i) The process X can be written as

Xt =∑

1≤j1,...,jn≤dXj1,...,jnt ej1 ⊗ . . .⊗ ejn ,

where the Xj1,...,jn are real-valued processes for any choice of j1, . . . , jn ∈ 1, . . . , d. Fixj, j1, . . . , jn ∈ 1, . . . , d. Using independence of the Brownian increments,

E

(m−1∑i=0

Xj1,...,jnti

Bjti,ti+1

)2 =

m−1∑i=0

E

((Xj1,...,jnti

)2 (Bjti,ti+1

)2)

=m−1∑i=0

∣∣∣Xj1,...,jnti

∣∣∣2L2

∣∣∣Bjti,ti+1

∣∣∣2L2

≤ maxi=0,...,m−1


∣∣∣2L2

m−1∑i=0

|ti+1 − ti|

≤ maxi=0,...,m−1

|Xti |2L2 |t− s| .

(ii) Fix j, k, j1, . . . , jn ∈ 1, . . . , d. Like for the Brownian increments, we know that Aj,kti,v and

25


Xj1,...,jnti

are independent and that E(Aj,kti,v) = 0 for all ti < v. Therefore,

E

(m−1∑i=0

Xj1,...,jnti

Aj,kti,ti+1

)2 =

m−1∑i=0

E

((Xj1,...,jnti

)2 (Aj,kti,ti+1

)2)

=m−1∑i=0


∣∣∣2L2

∣∣∣Aj,kti,ti+1

∣∣∣2L2

≤ c maxi=0,...,m−1


∣∣∣2L2

m−1∑i=0

|ti+1 − ti|2

≤ c |D| maxi=0,...,m−1


∣∣∣2L2

m−1∑i=0

|ti+1 − ti|

≤ c |D| maxi=0,...,m−1

|Xti |2L2 |t− s| .

DefineSN (B) : ∆→ TN

(Rd)

by

πn

(SN (B)s,t

)=

∫∆ns,tdB ⊗ . . .⊗ dB if n ∈ 1, . . . , N

1 if n = 0.

It is known that SN (B) is a multiplicative functional. In rough path terms, SN (B) is the stepN Lyons-lift of the enhanced brownian motion B.

SinceB andBD are Gaussian processes, the random variables πn

(SN (B)s,t

)and πn

(SN (BD)s,t

)are elements in the n-th (non-homogeneous) Wiener Chaos Cn (P). Note that for Z ∈ Cn (P)and q > 2,

|Z|L2 ≤ |Z|Lq ≤ |Z|L2 (n+ 1) (q − 1)n/2

(see e.g. [FV10b, Ch. 15, Sect. 3.1]). As a consequence, all Lq-norms are equivalent on Cn (P).In particular, for Z ∈ Cn (P) and W ∈ Cm (P),

|Z ⊗W |L2 ≤ c (n,m) |Z|L2 |W |L2 .

The next lemma contains the main work of this section.

Lemma 1.1.4. Let B be a Brownian motion on a probability space (Ω,F , (Ft) ,P). Let n ≤ Nand D = s = t0 < t1 < . . . < tm = t be any partition of any interval [s, t] ⊂ [0, 1]. Then thereis a constant c = c (n) such that∣∣∣πn (SN (BD)s,t − SN (B)s,t

)∣∣∣L2≤ c |D|

12 |t− s|

n−12 .

Remark 1.1.5. By the scaling property of the Brownian motion, it would have been enough toproof the lemma for (s, t) = (0, 1). Indeed, for arbitrary s < t,

πn

(SN (BD)s,t − SN (B)s,t

)D= |t− s|n/2 πn

(SN (BD)0,1 − SN (B)0,1

)where D =

0 = t0 < . . . < tm = 1

is defined by ti = ti−s

t−s for all i = 0, . . . ,m. Clearly,∣∣∣D∣∣∣ = |D||t−s| and hence∣∣∣πn (SN (BD)s,t − SN (B)s,t

)∣∣∣L2

= |t− s|n/2∣∣∣πn (SN (BD)0,1 − SN (B)0,1

)∣∣∣L2

≤ c |t− s|n/2∣∣∣D∣∣∣1/2

= c |D|1/2 |t− s|n−12

26


Proof. By induction over n. For the cases 1 and 2 and (s, t) = (0, 1), we use [FV10b, Prop.13.20] where we let 1

r ↓ 0. For general (s, t) ∈ ∆ the estimate follows from Remark 1.1.5.Suppose now that the statement is true for all n′ ∈ 1, . . . , n− 1. We have to show the

estimate for n with n ≥ 3. By Lemma 1.1.1, we know that∣∣∣πn (SN (BD)s,t − SN (B)s,t

)∣∣∣L2

≤

∣∣∣∣∣m−1∑i=0

πn

(SN (BD)ti,ti+1 − SN (B)ti,ti+1

)∣∣∣∣∣L2

+n−1∑l=1

∣∣∣∣∣m−1∑i=1

πn−l(SN (BD)s,ti)⊗ πl(SN (BD)ti,ti+1)

−πn−l(SN (B)s,ti)⊗ πl(SN (B)ti,ti+1)∣∣∣L2.

We claim that ∣∣∣∣∣m−1∑i=0

πn


)∣∣∣∣∣L2

≤ c0 |D|12 |t− s|

n−12 (1.3)

and ∣∣∣∣∣m−1∑i=1


−πn−l(SN (B)s,ti)⊗ πl(SN (B)ti,ti+1)∣∣∣L2

≤ cl |D|12 |t− s|

n−12 (1.4)

for l = 1, . . . , n− 1. Setting c = c0 + . . .+ cn−1 then gives us the desired result.We start with (1.3). Since n ≥ 3, we can use Lemma 1.1.2, (1.2) and an estimate of the

form |a− b| ≤ |a|+ |b| to see that∣∣∣∣∣m−1∑i=0

πn(SN (BD)ti,ti+1 − SN (B)ti,ti+1)

∣∣∣∣∣L2

≤m−1∑i=0

∣∣πn(SN (BD)ti,ti+1)∣∣L2 +

∣∣πn(SN (B)ti,ti+1)∣∣L2

≤ c (n)m−1∑i=0

|ti+1 − ti|n/2

≤ c |D|1/2m−1∑i=0

|ti+1 − ti|n−12

≤ c |D|1/2(m−1∑i=0

|ti+1 − ti|

)n−12

= c |D|1/2 |t− s|n−12

We used here the basic inequality

|a1|p + . . . |an|p ≤ (|a1|+ . . . |an|)p

27


which is true for p ≥ 1.Now we come to (1.4). First we consider the case l = 1. Then,

m−1∑i=1

πn−1

(SN (BD

)s,ti

)⊗BDti,ti+1 − πn−1 (SN (B)s,ti)⊗Bti,ti+1

=m−1∑i=1

πn−1

(SN (BD

s,ti)− SN (Bs,ti))⊗Bti,ti+1 .

Since πn−1

(SN (BD

s,ti)− SN (Bs,ti))

is Fti-measurable for all ti we can use Lemma 1.1.3 tosee that ∣∣∣∣∣

m−1∑i=1

πn−1

(SN (BD

s,ti)− SN (Bs,ti))⊗Bti,ti+1

∣∣∣∣∣L2

≤ c maxi=1,...,m−1

∣∣πn−1

(SN (BD

s,ti)− SN (Bs,ti))∣∣L2 |t− s|

1/2 .

For fixed ti, by induction hypothesis,∣∣πn−1

(SN (BD

s,ti)− SN (Bs,ti))∣∣L2 ≤ c

∣∣∣D∣∣∣1/2 |ti − s|n−22

≤ c |D|1/2 |t− s|n−22

where D = s = t0 < . . . < ti. Hence,∣∣∣∣∣m−1∑i=1

πn−1

(SN (BD

s,ti)− SN (Bs,ti))⊗Bti,ti+1

∣∣∣∣∣L2

≤ c |D| |t− s|n−12 .

Assume now l ∈ 2, . . . , n− 1. Using the equality a⊗b−a′⊗b′ = (a− a′)⊗b+a′⊗ (b−b′),we can decompose the sum (1.4) into two sums and obtain, applying the triangle inequality,∣∣∣∣∣

m−1∑i=1


−πn−l(SN (B)s,ti)⊗ πl(SN (B)ti,ti+1)∣∣∣L2

≤

∣∣∣∣∣m−1∑i=1

πn−l(SN (BD

s,ti)− SN (Bs,ti))⊗ πl(SN (BD)ti,ti+1)

∣∣∣∣∣L2

(1.5)

+

∣∣∣∣∣m−1∑i=1

πn−l (SN (B)s,ti)⊗ πl(SN (BD)ti,ti+1 − SN (B)ti,ti+1

)∣∣∣∣∣L2

(1.6)

We start estimating the sum (1.5). Using equivalence of Lq-norms yields∣∣∣∣∣m−1∑i=1

πn−l(SN (BD

s,ti)− SN (Bs,ti))⊗ πl(SN (BD)ti,ti+1)

∣∣∣∣∣L2

≤ c(n, l)

m−1∑i=1

∣∣πn−l (SN (BDs,ti)− SN (Bs,ti)

)∣∣L2

∣∣πl(SN (BD)ti,ti+1)∣∣L2 .

Now we use the induction hypothesis to see that for fixed ti,∣∣πn−l (SN (BDs,ti)− SN (Bs,ti)

)∣∣L2 ≤ c

∣∣∣D∣∣∣1/2 |ti − s|n−l−12

≤ c |D|1/2 |t− s|n−l−1

2

28


where D = s = t0 < t1 < . . . < ti. In (1.2) we have seen that∣∣πl (SN (BD)ti,ti+1

)∣∣ ≤ c |ti+1 − ti|l2

and since l2 ≥ 1,

m−1∑i=1

∣∣πn−l (SN (BDs,ti)− SN (Bs,ti)

)∣∣L2

∣∣πl(SN (BD)ti,ti+1)∣∣L2

≤ c |D|1/2 |t− s|n−l−1

2

m−1∑i=1

|ti+1 − ti|l2

≤ c |D|1/2 |t− s|n−l−1

2 |t− t1|l2

≤ c |D|1/2 |t− s|n−12 .

Now we come to the sum (1.6). We first consider the case l = 2. An easy computation shows

π2


)= S2

(BD)ti,ti+1

− S2(B)ti,ti+1

= −Ati,ti+1 .

Hence, ∣∣∣∣∣m−1∑i=1

πn−2 (SN (B)s,ti)⊗ π2


)∣∣∣∣∣L2

=

∣∣∣∣∣m−1∑i=1

πn−2 (SN (B)s,ti)⊗Ati,ti+1

∣∣∣∣∣L2

It is clear that the process u 7→ πn−2 (SN (B)s,u), u ≥ s, is adapted to the filtration (Fu) andwe can use Lemma 1.1.3 and Lemma 1.1.2 to see that∣∣∣∣∣

m−1∑i=1

πn−2 (SN (B)s,ti)⊗Ati,ti+1

∣∣∣∣∣L2

≤ c |D|1/2 maxi=1,...,m−1

|πn−2 (SN (Bs,ti))|L2 |t− s|1/2

≤ c |D|1/2 |tm−1 − s|n−22 |t− s|1/2

≤ c |D|1/2 |t− s|n−12

Finally we look at (1.6) for the cases l ∈ 3, . . . , n− 1. Equivalence of Lq-norms andLemma 1.1.2 show that∣∣∣∣∣

m−1∑i=1

πn−l (SN (B)s,ti)⊗ πl(SN (BD)ti,ti+1 − SN (B)ti,ti+1

)∣∣∣∣∣L2

≤ cm−1∑i=1

∣∣∣πn−l (SN (B)s,ti)∣∣∣L2

∣∣πl (SN (BD)ti,ti+1 − SN (B)ti,ti+1

)∣∣L2

≤ c

m−1∑i=1

|ti − s|n−l2(∣∣πl (SN (BD)ti,ti+1

)∣∣L2 +

∣∣πl (SN (B)ti,ti+1

)∣∣L2

)≤ c |t− s|

n−l2

m−1∑i=1

|ti+1 − ti|l2

≤ c |D|1/2 |t− s|n−l2

m−1∑i=1

|ti+1 − ti|l−12

≤ c |D|1/2 |t− s|n−12 .

This finishes the proof.

29


Lemma 1.1.6. Let B be a Brownian motion and D be any partition of [0, 1]. Take n ∈ N.Then there is a constant c depending only on n such that∣∣∣∣∣

∫∆ns,t

dBD ⊗ . . .⊗ dBD

∣∣∣∣∣L2

≤ c |t− s|n2

for any (s, t) ∈ ∆.

Proof. Assume first that s, t ∈ D. Define D to be the subpartition D = s = tj < . . . < tj+m = tof D. Clearly,

∣∣∣D∣∣∣ ≤ |t− s|. From Lemma 1.1.4 and Lemma 1.1.2 we get then∣∣∣∣∣∫

∆ns,t

dBD ⊗ . . .⊗ dBD

∣∣∣∣∣L2

≤

∣∣∣∣∣∫

∆ns,t

dBD ⊗ . . .⊗ dBD −∫

∆ns,t

dB ⊗ . . .⊗ dB

∣∣∣∣∣L2

+

∣∣∣∣∣∫

∆ns,t

dB ⊗ . . .⊗ dB

∣∣∣∣∣L2

≤ c1

∣∣∣D∣∣∣ 12 |t− s|n−12 + c2 |t− s|

n2

≤ (c1 + c2) |t− s|n2 .

Now assume that there are ti, ti+1 ∈ D such that ti ≤ s < t ≤ ti+1. Since on (ti, ti+1) we have

BD =Bti+1 −Btiti+1 − ti

D=

B1

(ti+1 − ti)1/2

we obtain then ∣∣∣∣∣∫

∆ns,t

dBD ⊗ . . .⊗ dBD

∣∣∣∣∣L2

=

∣∣B⊗n1

∣∣L2

(ti+1 − ti)n/2

∫∆ns,t

du1 · · · dun

≤∣∣B⊗n1

∣∣L2 |t− s|n

|t− s|n/2 n!

= c3 |t− s|n2 .

Finally, for ti−1 ≤ s ≤ ti < tj ≤ t ≤ tj+1, we use the identity

SN(BD)s,t

= SN(BD)s,ti⊗ SN

(BD)ti,tj⊗ SN

(BD)tj ,t

.

For n ≤ N , by our previous estimates,∣∣∣πn (SN (BD)s,t

)∣∣∣L2

=

∣∣∣∣∣∣∑

α+β+γ=n

πα

(SN(BD)s,ti

)⊗ πβ

(SN(BD)ti,tj

)⊗ πγ

(SN(BD)tj ,t

)∣∣∣∣∣∣L2

≤ c (n)∑

α+β+γ=n

∣∣∣πα (SN (BD)s,ti

)∣∣∣L2

∣∣∣πβ (SN (BD)ti,tj

)∣∣∣L2

∣∣∣πγ (SN (BD)tj ,t

)∣∣∣L2

≤ c∑

α+β+γ=n

|ti − s|α2 |tj − ti|

β2 |t− tj |

γ2

≤ c∑

α+β+γ=n

|t− s|α+β+γ

2 ≤ c |t− s|n2

and the proof is finished.

30


Theorem 1.1.7. Let D be any partition of [0, 1] and let N ∈ N. Then for all n ≤ N there is aconstant c = c (n) such that for all 1/r ∈ [0, 1/2] and (s, t) ∈ ∆∣∣∣πn (SN (BD)s,t − SN (B)s,t

)∣∣∣L2≤ c |D|1/2−1/r |t− s|

nr

Remark 1.1.8. This proposition is a generalization of [FV10b, Prop. 13.20] where the state-ment was shown for N = 2. One might wonder if all the work in this section could be avoidedby applying the Lipschitz property of the Lyons-lifting map SN (see e.g. [FV10b, Thm. 9.10]).

The reasoning would be as follows: By definition of ρ(n)α-Hol,∣∣∣πn (SN (BD)s,t − SN (B)s,t

)∣∣∣L2≤

∣∣∣ρ(n)α-Hol(SN (BD), SN (B))

∣∣∣L2|t− s|nα

≤∣∣ρα-Hol(SN (BD), SN (B))

∣∣L2 |t− s|nα

for every α ∈ (0, 1]. Lipschitz-continuity of SN tells us that for 1/3 < α ≤ 1/2,

ρα-Hol(SN (BD), SN (B)) ≤ cρα-Hol(S2(BD), S2(B))

and therefore∣∣∣πn (SN (BD)s,t − SN (B)s,t

)∣∣∣L2≤ c

∣∣ρα-Hol(S2(BD), S2(B))∣∣L2 |t− s|nα

≤ c |D|1/2−α′|t− s|nα

for α′ ∈ (α, 1/2] by [FV10b, Cor. 13.21]. The point we want to make here is that sinceα′ ∈ (1/3, 1/2], the optimal rate of convergence with this approach will only be arbitrary closeto 1/6. Theorem 1.1.7 on the other hand states that∣∣∣πn (SN (BD)s,t − SN (B)s,t

)∣∣∣L2≤ c |D|1/2−α |t− s|nα

with α ∈ (0, 1/2], hence we can choose α close to 0 to obtain a convergence rate close to 1/2.

Proof. Fix n ≤ N . Assume first that ti ≤ s < t ≤ ti+1 for ti, ti+1 ∈ D. Applying Lemma 1.1.2and 1.1.6 gives us∣∣∣πn (SN (BD)s,t − SN (B)s,t

)∣∣∣L2≤

∣∣πn (SN (BD)s,t)∣∣L2 +

∣∣∣πn (SN (B)s,t

)∣∣∣L2

≤ c1 |t− s|n2

= c1 |t− s|n−12 min |D| , |t− s|1/2 .

From Lemma 1.1.4, for s = ti < tj = t ∈ D,∣∣∣πn (SN (BD)s,t − SN (B)s,t

)∣∣∣L2≤ c2 |t− s|

n−12

∣∣∣D∣∣∣1/2= c2 |t− s|

n−12 min

∣∣∣D∣∣∣ , |t− s|1/2

≤ c2 |t− s|n−12 min |D| , |t− s|1/2

where D = s = ti < . . . < tj = t. Now assume that there are ti, tj ∈ D such that ti−1 ≤ s ≤ti < tj ≤ t ≤ tj+1. Since SN (BD) and SN (B) are multiplicative functionals,

SN (BD)s,t − SN (B)s,t =(SN (BD)s,ti − SN (B)s,ti

)⊗ SN (BD)ti,t

+SN (B)s,ti ⊗(SN (BD)ti,tj − SN (B)ti,tj

)⊗ SN (BD)tj ,t

+SN (B)s,tj ⊗(SN (BD)tj ,t − SN (B)tj ,t

)31


Now we project this down on the n-th level and use the previous estimates. Note that for allu < v,

π0

(SN (BD)u,v − SN (B)u,v

)= 1− 1 = 0

and for ti < tj ∈ D,

π1

(SN (BD)ti,tj − SN (B)ti,tj

)= Bti,tj −Bti,tj = 0.

Hence ∣∣∣πn ((SN (BD)s,ti − SN (B)s,ti

)⊗ SN (BD)ti,t

)∣∣∣L2

=

∣∣∣∣∣n∑l=1

πl

(SN (BD)s,ti − SN (B)s,ti

)⊗ πn−l

(SN (BD)ti,t

)∣∣∣∣∣L2

≤ c(n)n∑l=1

∣∣∣πl (SN (BD)s,ti − SN (B)s,ti

)∣∣∣L2

∣∣πn−l (SN (BD)ti,t)∣∣L2

≤ cn∑l=1

|ti − s|l−12 min |D| , |ti − s|1/2 |t− ti|

n−l2

≤ cmin |D| , |ti − s|1/2n∑l=1

|t− s|l−1+n−l

2

≤ c3 min |D| , |t− s|1/2 |t− s|n−12 .

In the same way one obtains∣∣∣πn (SN (B)s,tj ⊗(SN (BD)tj ,t − SN (B)tj ,t

))∣∣∣L2≤ c4 min |D| , |t− s|1/2 |t− s|

n−12 .

and ∣∣∣πn (SN (B)s,ti ⊗(SN (BD)ti,tj − SN (B)ti,tj

)⊗ SN (BD)tj ,t

)∣∣∣L2

≤ c5 min |D| , |t− s|1/2 |t− s|n−12 .

Using the triangle inequality, we end up with∣∣∣πn (SN (BD)s,t − SN (B)s,t

)∣∣∣L2≤ (c3 + c4 + c5) min |D| , |t− s|1/2 |t− s|

n−12 .

We have shown that this estimate holds for all s < t. Assume 2/r ∈ (0, 1). By geometricinterpolation,∣∣∣πn (SN (BD)s,t − SN (B)s,t

)∣∣∣L2≤ c |t− s|

n−12

(|D|1−2/r |t− s|2/r

)1/2

≤ c |D|1/2−1/r |t− s|n−1r |t− s|

1r

= c |D|1/2−1/r |t− s|nr .

Since the constant c does not depend on r, this estimate also holds for 1/r ∈ 0, 1/2.

Corollary 1.1.9. Let 0 ≤ α < 1/2. Then, for every η ∈ (0, 1/2 − α), there is a constantc = c(α, η,N) such that for all q ∈ [1,∞),∣∣ρα-Hol(SN (BD), SN (B))

∣∣Lq≤ cqN/2 |D|η

32


Proof. Set 1/r := 1/2 − η. Note that with this choice, α < 1/r < 1/2. By Theorem 1.1.7, weknow that for all n = 1, . . . , N∣∣∣πn (SN (BD)s,t − SN (B)s,t

)∣∣∣L2≤ c1 |D|1/2−1/r |t− s|

nr .

Lemma 1.1.2 and 1.1.6 show that

|πn (SN (B)s,t)|L2 ≤ c2 |t− s|nr ,∣∣πn (SN (BD)s,t

)∣∣L2 ≤ c3 |t− s|

nr .

Applying [FV10b, Prop. 15.24], we obtain∣∣∣ρ(n)α-Hol(SN (BD), SN (B))

∣∣∣Lq≤ c(n, α, η,N)q

n2 |D|1/2−1/r

≤ c (α, η,N) qN/2 |D|η ,

therefore, ∣∣ρα-Hol(SN (BD), SN (B))∣∣Lq≤ cqN/2 |D|η .

The next theorem states the main result of this section.

Theorem 1.1.10. Let B be a Brownian motion on a probability space(

Ω,F , (Ft)t∈[0,1] ,P)

.

Assume that we have a sequence of partitions (Dn)n∈N of [0, 1] such that the sequence (|Dn|)n∈Nof real numbers is contained in

⋃q≥1 l

q. Let N ∈ N, 0 ≤ α < 1/2 and η ∈ (0, 1/2 − α). Thenthere is a (almost surely finite) random variable C, F measurable, depending also on α, η andN , such that

ρα-Hol(SN (BDn), SN (B)) ≤C |Dn|η a.s.

for all n ∈ N.

Remark 1.1.11. We shall apply this theorem with N = [1/α] which makes ρα-Hol a rough pathmetric and let α ↓ 0 to obtain optimal rates of convergence.

Remark 1.1.12. The typical example of a sequence of partitions (Dn)n∈N satisfying the condi-tion in the theorem are the dyadic partitions. Another example would be the uniform partitions

Dn =

0 <

1

n<

2

n< . . . <

n− 1

n< 1

since for q > 1,

|(|Dn|)|qlq =∞∑n=1

(1

n

)q<∞.

Proof. Choose η′ such that η < η′ < 1/2 − α and define ε := η′ − η > 0. Applying Corollary(1.1.9) with η′ gives us ∣∣∣∣ρα-Hol(SN (BDn), SN (B))

|Dn|η∣∣∣∣Lq≤ cqN/2 |Dn|ε .

for every q ≥ 1. Using the Markov inequality shows that for every δ > 0,

∞∑n=1

P

[ρα-Hol(SN (BDn), SN (B))

|Dn|η≥ δ]≤ 1

δq

∞∑n=1

∣∣∣∣ρα-Hol(SN (BDn), SN (B))

|Dn|η∣∣∣∣qLq

≤ c

∞∑n=1

|Dn|qε .

33


From the assumptions on (Dn) we can choose q big enough such that the series converge. WithBorell-Cantelli we conclude that

ρα-Hol(SN (BDn), SN (B))

|Dn|η→ 0 a.s.

for n→∞. We set

C := supn

ρα-Hol(SN (BDn), SN (B))

|Dn|η

which is finite almost surely. Since C is the supremum of F-measurable random variables, it isitself F-measurable.

34

2

Convergence rates for the fullGaussian rough paths

Recall that rough path theory [Lyo98, LQ02, FV10b] is a general framework that allows toestablish existence, uniqueness and stability of differential equations driven by multi-dimensionalcontinuous signals x : [0, T ] → Rd of low regularity. Formally, a rough differential equation(RDE) is of the form

dyt =d∑i=1

Vi (yt) dxit ≡ V (yt) dxt; y0 ∈ Re (2.1)

where (Vi)i=1,...,d is a family of vector fields in Re. When x has finite p-variation, p < 2, suchdifferential equations can be handled by Young integration theory. Of course, this point of viewdoes not allow to handle differential equations driven by Brownian motion, indeed

supD⊂[0,T ]

∑ti∈D

∣∣Bti+1 −Bti∣∣2 = +∞ a.s.,

leave alone differential equations driven by stochastic processes with less sample path regularitythan Brownian motion (such as fractional Brownian motion (fBM) with Hurst parameter H <1/2). Lyons’ key insight was that low regularity of x, say p-variation or 1/p-Holder for somep ∈ [1,∞), can be compensated by including ”enough” higher order information of x such asall increments

xns,t ≡∫s<t1<···<tn<t

dxt1 ⊗ . . .⊗ dxtn (2.2)

≡∑

1≤i1,...,in≤d

(∫s<t1<···<tn<t

dxi1t1 . . . dxintn

)ei1 ⊗ . . .⊗ ein ∈

(Rd)⊗n

(2.3)

where ”enough” means n ≤ [p] (e1, . . . , ed denotes just the usual Euclidean basis in Rd here).Subject to some generalized p-variation (or 1/p-Holder) regularity, the ensemble

(x1, . . . ,x[p]

)then constitutes what is known as a rough path.1 In particular, no higher order information isnecessary in the Young case; whereas the regime relevant for Brownian motion requires secondorder - or level 2 - information (”Levy’s area”), and so on. Note that the iterated integral on the

1A basic theorem of rough path theory asserts that further iterated integrals up to any level N ≥ [p], i.e.

SN (x) := (xn : n ∈ 1, . . . , N)

are then deterministically determined and the map x 7→ SN (x), known as Lyons lift, is continuous in rough pathmetrics.

Rates Gaussian rough paths

r.h.s. of (2.2) is not - in general - a well-defined Riemann-Stieltjes integral. Instead one typicallyproceeds by mollification - given a multi-dimensional sample path x = X (ω), consider piecewiselinear approximations or convolution with a smooth kernel, compute the iterated integrals andthen pass, if possible, to a limit in probability. Following this strategy one can often constructa ”canonical” enhancement of some stochastic process to a (random) rough path. Stochasticintegration and differential equations are then discussed in a (rough) pathwise fashion; even inthe complete absence of a semi-martingale structure.

It should be emphasized that rough path theory was - from the very beginning - closelyrelated to higher order Euler schemes. Let D = 0 = t0 < . . . < t#D−1 = 1 be a partition of

the unit interval.2 Considering the solution y of (2.1), the step-N Euler approximation yEulerN ;D

is given by

yEulerN ;D0 = y0

yEulerN ;Dtj+1

= yEulerN ;Dtj

+ Vi

(yEulerN ;Dtj

)xitj ,tj+1

+ Vi1Vi2(yEulerN ;Dtj

)xi1,i2tj ,tj+1

+ . . .+ Vi1 . . .V iN−1ViN

(yEulerN ;Dtj

)xi1,...,iNtj ,tj+1

at the points tj ∈ D where we use the Einstein summation convention, Vi stands for the

differential operator∑e

k=1 Vki ∂xk and xi1,...,ins,t =

∫s<t1<···<tn<t dx

i1t1. . . dxintn . An extension of the

work of A.M. Davie (cf. [Dav07], [FV10b]) shows that the step-N Euler scheme3 for an RDE

driven by a 1/p-Holder rough path with step size 1/k (i.e. D = Dk =jk : j = 0, . . . , k

) and

N ≥ [p] will converge with rate O(

1k

)(N+1)/p−1. Of course, in a probabilistic context, simulation

of the iterated (stochastic) integrals xntj ,tj+1is not an easy matter. A natural simplification of

the step-N Euler scheme thus amounts to replace in each stepxntj ,tj+1

: n ∈ 1, . . . , N↔

1

n!

(x1tj ,tj+1

)⊗n: n ∈ 1, . . . , N

which leads to the simplified step-N Euler scheme

ysEulerN ;D0 = y0

ysEulerN ;Dtj+1

= ysEulerN ;Dtj

+ Vi

(ysEulerN ;Dtj

)xitj ,tj+1

+1

2Vi1Vi2

(ysEulerN ;Dtj

)xi1tj ,tj+1

xi2tj ,tj+1

+ . . .+1

N !Vi1 . . .V iN−1ViN

(ysEulerN ;Dtj

)xi1tj ,tj+1

. . .xiNtj ,tj+1.

Since x1tj ,tj+1

= Xtj ,tj+1 (ω) = Xtj+1 (ω) − Xtj (ω) this is precisely the effect in replacing theunderlying sample path segment of X by its piecewise linear approximation, i.e.

Xt (ω) : t ∈ [tj , tj+1] ↔Xtj (ω) +

t− tjtj+1 − tj

Xtj ,tj+1 (ω) : t ∈ [tj , tj+1]

.

Therefore, as pointed out in [DNT12] in the level N = 2 Holder rough path context, it isimmediate that a Wong-Zakai type result, i.e. a.s. convergence of y(k) → y for k → ∞ wherey(k) solves

dy(k)t = V

(y

(k)t

)dx

(k)t ; y

(k)0 = y0 ∈ Re

and x(k) is the piecewise linear approximation of x at the points (tj)kj=0 = Dk, i.e.

x(k)t = xtj +

t− tjtj+1 − tj

xtj ,tj+1 if t ∈ [tj , tj+1] , tj ∈ Dk,

2A general time horizon [0, T ] is handled by trivial reparametrization of time.3... which one would call Milstein scheme when N = 2 ...

36

leads to the convergence of the simplified (and implementable!) step-N Euler scheme.

While Wong-Zakai type results in rough path metrics are available for large classes of stochas-tic processes [FV10b, Chapter 13, 14, 15, 16] our focus here is on Gaussian processes which canbe enhanced to rough paths. This problem was first discussed in [CQ02] where it was shownin particular that piecewise linear approximation to fBM are convergent in p-variation roughpath metric if and only if H > 1/4. A practical (and essentially sharp) structural conditionfor the covariance, namely finite ρ-variation based on rectangular increments for some ρ < 2 ofthe underlying Gaussian process was given in [FV10a] and allowed for a unified and detailedanalysis of the resulting class of Gaussian rough paths. This framework has since proven usefulin a variety of different applications ranging from non-Markovian Hormander theory [CF10] tonon-linear PDEs perturbed by space-time white-noise [Hai11]. Of course, fractional Brownianmotion can also be handled in this framework (for H > 1/4) and we shall make no attempt tosurvey its numerous applications in engineering, finance and other fields.

Before describing our main result, let us recall in more detail some aspects of Gaussian roughpath theory (e.g. [FV10a], [FV10b, Chapter 15], [FV11]). The basic object is a centred, contin-uous Gaussian process with sample paths X (ω) =

(X1 (ω) , . . . , Xd (ω)

): [0, 1]→ Rd where Xi

and Xj are independent for i 6= j. The law of this process is determined by RX : [0, 1]2 → Rd×d,the covariance function, given by

RX (s, t) = diag(E(X1sX

1t

), . . . , E

(XdsX

dt

)).

The main result in this context (see e.g. [FV10b, Theorem 15.33], [FV11]) now asserts

that if there exists ρ < 2 such that Vρ

(RX , [0, 1]2

)<∞ then X lifts to an enhanced Gaussian

process X with sample paths in the p-variation rough path space C0,p−var ([0, 1] , G[p](Rd))

, anyp ∈ (2ρ, 4). This lift is ”natural” in the sense that for a large class of smooth approximationsX(k) of X (say piecewise linear, mollifier, Karhunen-Loeve) the corresponding iterated integralsof X(k) converge (in probability) to X with respect to the p-variation rough path metric. (Werecall from [FV10b] that ρp-var, the so-called inhomogeneous p-variation metric for GN

(Rd)-

valued paths, is called p-variation rough path metric when [p] = N ; the Ito-Lyons map enjoyslocal Lipschitz regularity in this p-variation rough path metric.) Moreover, this condition issharp; indeed fBM falls into this framework with ρ = 1/ (2H) and we known that piecewise-linear approximations to Levy’s area diverge when H = 1/4.

Our main result (cf. Theorem 2.4.1), when applied to (mesh-size 1/k) piecewise linearapproximations X(k) of X, reads as follows.

Theorem 2.0.1. Let X =(X1, . . . , Xd

): [0, 1] → Rd be a centred Gaussian process on a

probability space (Ω,F , P ) with continuous sample paths where Xi and Xj are independentfor i 6= j. Assume that the covariance RX has finite ρ-variation for ρ ∈ [1, 2) and K ≥Vρ

(RX , [0, 1]2

). Then there is an enhanced Gaussian process X with sample paths a.s. in

C0,p−var ([0, 1] , G[p](Rd))

for any p ∈ (2ρ, 4) and∣∣∣ρp−var (S[p]

(X(k)

),X)∣∣∣Lr→ 0

for k → ∞ and every r ≥ 1 (|·|Lr denotes just the usual Lr (P )-norm for real valued randomvariables here). Moreover, for any γ > ρ such that 1

γ + 1ρ > 1 and any q > 2γ and N ∈ N there

is a constant C = C (q, ρ, γ,K,N) such that∣∣∣ρq−var (SN (X(k)), SN (X)

)∣∣∣Lr≤ CrN/2 sup

0≤t≤1

∣∣∣X(k)t −Xt

∣∣∣1− ργL2

holds for every k ∈ N.

37


As an immediate consequence we obtain (essentially) sharp a.s. convergence rates for Wong-Zakai approximations and the simplified step-3 Euler scheme.

Corollary 2.0.2. Consider a RDE with C∞-bounded vector fields driven by a Gaussian Holderrough path X. Then mesh-size 1/k Wong-Zakai approximations (i.e. solutions of ODEs drivenby X(k)) converge uniformly with a.s. rate k−(1/ρ−1/2−ε), any ε > 0, to the RDE solution. Thesame rate is valid for the simplified (and implementable) step-3 Euler scheme.

Proof. See Corollary 2.4.5 and Corollary 2.4.7.

Several remarks are in order.

• Rough path analysis usually dictates thatN = 2 (resp. N = 3) levels need to be consideredwhen ρ ∈ [1, 3/2) resp. ρ ∈ [3/2, 2). Interestingly, the situation for the Wong-Zakai erroris quite different here - referring to Theorem 2.0.1, when ρ = 1 we can and will take γarbitrarily large in order to obtain the optimal convergence rate. Since ρq−var is a roughpath metric only in the case N = [q] ≥ [2γ], we see that we need to consider all levels Nwhich is what Theorem 2.0.1 allows us to do. On the other hand, as ρ approaches 2, thereis not so much room left for taking γ > ρ. Even so, we can always find γ with [γ] = 2such that 1/γ + 1/ρ > 1. Picking q > 2γ small enough shows that we need N = [q] = 4.

• The assumption of C∞-bounded vector fields in the corollary was for simplicity only. Inthe proof we employ local Lipschitz continuity of the Ito-Lyons map for q-variation roughpaths (involving N = [q] levels). As is well-known, this requires Lipq+ε-regularity of thevector fields4. Curiously again, we need C∞-bounded vector fields when ρ = 1 but onlyLip4+ε as ρ approaches the critical value 2.

• Brownian motion falls in this framework with ρ = 1. While the a.s. (Wong-Zakai)rate k−(1/2−ε) is part of the folklore of the subject (e.g. [GS06]) the C∞-boundednessassumption appears unnecessarily strong. Our explanation here is that our rates areuniversal (i.e. valid away from one universal null-set, not dependent on starting points,coefficients etc). In particular, the (Wong-Zakai) rates are valid on the level of stochasticflows of diffeomorphisms; we previously discussed these issues in the Brownian context in[FR11].

• A surprising aspect appears in the proof of Theorem 2.0.1. The strategy is to give sharpestimates for the levels n = 1, . . . , 4 first, then performing an induction similar to the oneused in Lyon’s Extension Theorem ([Lyo98]) for the higher levels. This is in contrast tothe usual considerations of level 1 to 3 only (without level 4!) which is typical for Gaussianrough paths. (Recall that we deal with Gaussian processes which have sample paths offinite p-variation, p ∈ (2ρ, 4), hence [p] ≤ 3 which indicates that we would need to controlthe first 3 levels only before using the Extension Theorem.)

• Although Theorem 2.0.1 was stated here for (step-size 1/k) piecewise linear approxima-tions

X(k)

, the estimate holds in great generality for (Gaussian) approximations whose

covariance satisfies a uniform ρ-variation bound. The statements of Theorem 2.4.1 andTheorem 2.4.2 reflect this generality.

• Wong-Zakai rates for the Brownian rough path (level 2) were first discussed in [HN09].They prove that Wong-Zakai approximations converge (in γ-Holder metric) with ratek−(1/2−γ−ε) (in fact, a logarithmic sharpening thereof without ε) provided γ ∈ (1/3, 1/2).This restriction on γ is serious (for they fully rely on ”level 2” rough path theory); inparticular, the best ”uniform” Wong-Zakai convergence rate implied is k−(1/2−1/3−ε) =k−(1/6−ε) leaving a significant gap to the well-known Brownian a.s. Wong-Zakai rate.

4...in the sense of E. Stein; cf. [LQ02, FV10b] for instance.

38

Iterated integrals and the shuffle algebra

• Wong-Zakai (and Milstein) rates for the fractional Brownian rough path (level 2 only,Hurst parameter H > 1/3) were first discussed in [DNT12]. They prove that Wong-Zakai approximations converge (in γ-Holder metric) with rate k−(H−γ−ε) (again, in fact, alogarithmic sharpening thereof without ε) provided γ ∈ (1/3, H). Again, the restriction onγ is serious and the best ”uniform” Wong-Zakai convergence rate - and the resulting ratefor the Milstein scheme - is k−(H−1/3−ε). This should be compared to the rate k−(2H−1/2−ε)

obtained from our corollary. In fact, this rate was conjectured in [DNT12] and is sharp asmay be seen from a precise result concerning Levy’s stochastic area for fBM, see [NTU10].

The remainder of this chapter is structured as follows: Section 2.1 recalls the connectionbetween the shuffle algebra and iterated integrals. In particular, we will use the shuffle structureto see that in order to show the desired estimates, we can concentrate on some iterated integralswhich somehow generate all the others. Our main tool for showing L2 estimates on the lowerlevels is multidimensional Young integration which we present in Section 2.2. The main work,namely showing the desired L2-estimates for the difference of high-order iterated integrals, isdone in Section 2.3. After some preliminary Lemmas in Subsection 2.3.1, we show the estimatesfor the lower levels, namely for n = 1, 2, 3, 4 in Subsection 2.3.2 , then give an induction argumentin Subsection 2.3.3 for the higher levels n > 4. Section 2.4 contains our main result, namelysharp a.s. convergence rates for a class of Wong-Zakai approximations, including piecewise-linear and mollifier approximations. We further show in Subsection 2.4.3 how to use theseresults in order to obtain sharp convergence rates for the simplified Euler scheme.

As already mentioned, we can restrict ourselves to the case [0, T ] = [0, 1] in the followingchapter.

2.1 Iterated integrals and the shuffle algebra

Let x =(x1, . . . , xd

): [0, 1] → Rd be a path of finite variation. Forming finite linear combina-

tions of iterated integrals of the form∫∆n

0,1

dxi1 . . . dxin , i1, . . . , in ∈ 1, . . . , d , n ∈ N

defines a vector space over R. In this section, we will see that this vector space is also analgebra where the product is given simply by taking the usual multiplication. Moreover, we willdescribe precisely how the product of two iterated integrals looks like.

1.1 The shuffle algebra

Let A be a set which we will call from now on the alphabet. In the following, we will onlyconsider the finite alphabet A = a, b, . . . = a1, a2, . . . , ad = 1, . . . , d. We denote by A∗ theset of words composed by the letters of A, hence w = ai1ai2 . . . ain , aij ∈ A. The empty word isdenoted by e. A+ is the set of non-empty words. The length of the word is denoted by |w| and|w|a denotes the number of occurrences of the letter a. We denote by R 〈A〉 the vector spaceof noncommutative polynomials on A over R, hence every P ∈ R 〈A〉 is a linear combination ofwords in A∗ with coefficients in R. (P,w) denotes the coefficient in P of the word w. Henceevery polynomial P can be written as

P =∑w∈A∗

(P,w)w

and the sum is finite since the (P,w) are non-zero only for a finite set of words w. We definethe degree of P as

deg (P ) = max |w| ; (P,w) 6= 0 .

39


A polynomial is called homogeneous if all monomials have the same degree. We want to definea product on R 〈A〉. Since a polynomial is determined by its coefficients on each word, we candefine the product PQ of P and Q by

(PQ,w) =∑w=uv

(P, u)(Q, v).

Note that this definition coincides with the usual multiplication in a (noncommutative) polyno-mial ring. We call this product the concatenation product and the algebra R 〈A〉 endowed withthis product the concatenation algebra.

There is another product on R 〈A〉 which will be of special interest for us. We need some no-tation first. Given a word w = ai1ai2 . . . ain and a subsequence U = (j1, j2, . . . , jk) of (i1, . . . , in),we denote by w(U) the word aj1aj2 . . . ajk and we call w(U) a subword of w. If w, u, v are words

and if w has length n, we denote by

(w

u v

)the number of subsequences U of (1, . . . , n) such

that w(U) = u and w(U c) = v.

Definition 2.1.1. The (homogeneous) polynomial

u ∗ v =∑w∈A∗

(w

u v

)w

is called the shuffle product of u and v. By linearity we extend it to a product on R 〈A〉.

In order to proof our main result, we want to use some sort of induction over the length ofthe words. Therefore, the following definition will be useful.

Definition 2.1.2. If U is a set of words of the same length, we call a subset w1, . . . , wk ofU a generating set for U if for every word w ∈ U there is a polynomial R and real numbersλ1, . . . , λk such that

w =

k∑j=1

λjwj +R

where R is of the form R =∑

u,v∈A+ µu,vu ∗ v for real numbers µu,v.

Definition 2.1.3. We say that a word w is composed by an11 , . . . , andd if w ∈ a1, . . . , ad∗ and

|w|ai = ni for i = 1, . . . , d, hence every letter appears in the word with the given multiplicity.

The aim now is to find a (possibly small) generating set for the set of all words composedby some given letters. The next definition introduces a special class of words which will beimportant for us.

Definition 2.1.4. Let A be totally ordered and put on A∗ the alphabetical order. If w is a wordsuch that whenever w = uv for u, v ∈ A+ one has u < v, then w is called a Lyndon word.

Proposition 2.1.5. (i) For the set words composed by a, a, b a generating set is given byaab.

(ii) For the set words composed by a, a, a, b a generating set is given by aaab.

(iii) For the set words composed by a, a, b, b a generating set is given by aabb.

(iv) For the set words composed by a, a, b, c a generating set is given by aabc, aacb, baac.

40

Iterated integrals and the shuffle algebra

Proof. Consider the alphabet A = a, b, c. We choose the order a < b < c. A generaltheorem states that every word w has a unique decreasing factorization into Lyndon words, i.e.w = li11 . . . likk where l1 > . . . > lk are Lyndon words and i1, . . . , ik ≥ 1 (see [Reu93, Theorem5.1 and Corollary 4.7]), and the formula

1

i1! . . . ik!l∗i11 ∗ . . . ∗ l∗ikk = w +

∑u<w

αuu

holds, where αu are some natural integers (see again [Reu93, Theorem 6.1]). By repeatedlyapplying this formula for the words in the sum on the right hand side, it follows that a generatingset for each of the sets in (1) to (4) is given exactly by the Lyndon words composed by theseletters. One can easily show that indeed aab, aaab and aabb are the only Lyndon words composedby the corresponding letters. The Lyndon words composed by a, a, b, c are aabc, abac, aacbwhich therefore is a generating set for words composed by a, a, b, c. From the shuffle identity

abac = baac+ aabc+ aacb− b ∗ aac

it follows that also aabc, aacb, baac generates this set.

1.2 The connection to iterated integrals

Let x = (x1, . . . , xd) : [0, 1] → Rd be a path of finite variation and fix s < t ∈ [0, 1]. For aword w = (ai1 . . . ain) ∈ A∗, A = 1, . . . , d we define

xw =

∫∆ns,tdxi1 . . . dxin if w ∈ A+

1 if w = e.

Let (R 〈A〉 ,+, ∗) be the shuffle algebra over the alphabet A. We define a map Φ: R 〈A〉 → Rby Φ (w) = xws,t and extend it linearly to polynomials P ∈ R 〈A〉. The key observation is thefollowing:

Theorem 2.1.6. Φ is an algebra homomorphism from the shuffle algebra (R 〈A〉 ,+, ∗) to(R,+, ·).

Proof. [Reu93], Corollary 3.5.

In the following, X will denote a Gaussian process as in Theorem 2.0.1 and X denotes thenatural Gaussian rough path. We will need the following Proposition:

Proposition 2.1.7. Let X be as in Theorem 2.0.1 and assume that ω controls the ρ-variationof the covariance of X, ρ ∈ [1, 2). Then for every n ∈ N there is a constant C (n) = C (n, ρ)such that ∣∣Xn

s,t

∣∣L2 ≤ C (n)ω

([s, t]2

) n2ρ

for any s < t.

Proof. For n = 1, 2, 3 this is proven in [FV10b, Proposition 15.28]. For n ≥ 4 and fixeds < t, we set Xτ := 1

ω([s,t]2)12ρXs+τ(t−s). Then

∣∣RX ∣∣ρρ−var;[0,1]≤ 1 =: K and by the standard

(deterministic) estimates for the Lyons lift,∣∣Xns,t

∣∣1/nω(

[s, t]2) 1

2ρ

≤ c1

∥∥∥Sn (X)∥∥∥

p−var;[0,1]≤ c2 (n, p)

∥∥∥X∥∥∥p−var;[0,1]

41


for any p ∈ (2ρ, 4). Now we take the L2-norm on both sides. From [FV10b, Theorem 15.33]

we know that

∣∣∣∣∥∥∥X∥∥∥p−var;[0,1]

∣∣∣∣L2

is bounded by a constant only depending on p, ρ and K which

shows the claim.Alternatively (and more in the spirit of the forthcoming arguments), one performs an in-

duction similar (but easier) as in the proof of Proposition 2.3.21.

The next proposition shows that we can restrict ourselves in showing the desired estimatesonly for the iterated integrals which generate the others.

Proposition 2.1.8. Let (X,Y ) =(X1, Y 1, . . . , Xd, Y d

)be a Gaussian process on [0, 1] with

paths of finite variation. Let A = 1, . . . , d be the alphabet, let U be a set of words of lengthn and V = w1, . . . , wk be a generating set for U . Let ω be a control, ρ, γ ≥ 1 constants ands < t ∈ [0, 1]. Assume that there are constants C = C (|w|) such that∣∣Xw

s,t

∣∣L2 ≤ C (|w|)ω (s, t)

|w|2ρ and

∣∣Yws,t

∣∣L2 ≤ C (|w|)ω (s, t)

|w|2ρ

holds for every word w ∈ A∗ with |w| ≤ n− 1. Assume also that for some ε > 0∣∣Xws,t −Yw

s,t

∣∣L2 ≤ C (|w|) εω (s, t)

12γ ω (s, t)

|w|−12ρ

holds for every word w with |w| ≤ n− 1 and w ∈ V . Then there is a constant C which dependson the constants C, on n and on d such that∣∣Xw

s,t −Yws,t

∣∣L2 ≤ Cεω (s, t)

12γ ω (s, t)

n−12ρ

holds for every w ∈ U .

Remark 2.1.9. We could account for the factor ω (s, t)12γ in ε here but the present form is how

we shall use this proposition later on.

Proof. Consider a copy A of A. If a ∈ A, we denote by a the corresponding letter in A. Ifw = ai1 . . . ain ∈ A∗, we define w = ai1 . . . ain ∈ A∗ and in the same way we define P ∈ R

⟨A⟩

for P ∈ R 〈A〉. Now we consider R⟨A∪A

⟩equipped with the usual shuffle product. Define

Ψ: R⟨A∪A

⟩→ R by

Ψ (w) =

∫∆ns,t

dZbi1 . . . dZbin

for a word w = bi1 . . . bin where

Zbj =

Xaj for bj = ajY aj for bj = aj

and extend this definition linearly. By Theorem 2.1.6, we know that Ψ is an algebra homo-morphism. Take w ∈ U . By assumption, we know that there is a vector λ = (λ1, . . . , λk) suchthat

w − w =

k∑j=1

λj (wj − wj) +R− R

where R is of the form R =∑

u,v∈A+,|u|+|v|=n µu,v u ∗ v with real numbers µu,v. Applying Ψ and

taking the L2 norm yields

∣∣Xws,t −Yw

s,t

∣∣L2 ≤

k∑l=1

|λj |∣∣Xwj

s,t −Ywjs,t

∣∣L2 +

∣∣Ψ (R− R)∣∣L2

≤ c1εω (s, t)12γ ω (s, t)

n−12ρ +

∣∣Ψ (R− R)∣∣L2 .

42

Multidimensional Young-integration and grid-controls

Now,

R− R =∑u,v

µu,v (u ∗ v − u ∗ v) =∑u,v

µu,v (u− u) ∗ v + µu,vu ∗ (v − v) .

Applying Ψ and taking the L2 norm gives then∣∣Ψ (R− R)∣∣L2 ≤

∑u,v

|µu,v|∣∣(Xu

s,t −Yus,t

)Xvs,t

∣∣L2 + |µu,v|

∣∣Yus,t

(Xvs,t −Yv

s,t

)∣∣L2

≤∑u,v

c2

(∣∣Xus,t −Yu

s,t

∣∣L2

∣∣Xvs,t

∣∣L2 +

∣∣Yus,t

∣∣L2

∣∣Xvs,t −Yv

s,t

∣∣L2

)≤

∑u,v

c3εω (s, t)12γ ω (s, t)

|v|+|u|−12ρ

≤ c4εω (s, t)12γ ω (s, t)

n−12ρ

where we used equivalence of Lq-norms in the Wiener Chaos (cf. [FV10b, Proposition 15.19and Theorem D.8]). Putting all together shows the assertion.

2.2 Multidimensional Young-integration and grid-controls

Let f : [0, 1]n → R be a continuous function. If s1 < t1, . . . , sn < tn and u1, . . . , un are elementsin [0, 1], we make the following recursive definition:

f

s1, t1u2...un

: = f

t1u2...un

− f

s1

u2...un

and

f

s1, t1...

sk−1, tk−1

sk, tkuk+1

...un

: = f

s1, t1...

sk−1, tk−1

tkuk+1

...un

− f

s1, t1...

sk−1, tk−1

skuk+1

...un

.

We will also use the simpler notation

f (R) = f

s1, t1...

sn, tn

for the rectangle R = [s1, t1] × . . . × [sn, tn] ⊂ [0, 1]n. Note that for n = 2 this is consistent

with our initial definition of f

(s1, t1s2, t2

). If f, g : [0, 1]n → R are continuous functions, the

n-dimensional Young-integral is defined by∫[s1,t1]×...×[sn,tn]

f (x1, . . . , xn) dg (x1, . . . , xn)

: = lim|D1|,...,|Dn|→0

∑(t1i1

)⊂D1

...(tnin)⊂Dn

f(t1i1 , . . . , t

nin

)g

t1i1 , t1i1+1...

tnin , tnin+1

43


if this limit exists. Take p ≥ 1. The n-dimensional p-variation of f is defined by

Vp (f, [s1, t1]× . . .× [sn, tn]) =

sup

D1⊂[s1,t1]

...Dn⊂[sn,tn]

∑(t1i1

)⊂D1

...(tnin)⊂Dn

∣∣∣∣∣∣∣f t1i1 , t

1i1+1...

tnin , tnin+1

∣∣∣∣∣∣∣p

1/p

and if Vp (f, [0, 1]n) <∞ we say that f has finite (n-dimensional) p-variation. The fundamentaltheorem is the following:

Theorem 2.2.1. Assume that f has finite p-variation and g finite q-variation where 1p + 1

q > 1.Then the joint Young-integral below exists and there is a constant C = C (p, q) such that∣∣∣∣∣∣∣

∫[s1,t1]×...×[sn,tn]

f

s1, u1...

sn, un

dg (u1, . . . , un)

∣∣∣∣∣∣∣≤ CVp (f, [s1, t1]× . . .× [sn, tn])Vq (g, [s1, t1]× . . .× [sn, tn]) .

Proof. [Tow02], Theorem 1.2 (c).

We will mainly consider the case n = 2, but we will also need n = 3 and 4 later on. Inparticular, the discussion of level n = 4 will require us to work with 4D grid control functionswhich we now introduce. With no extra complication we make the following general definition.

Definition 2.2.2 (n-dimensional grid control). A map ω : ∆× . . .×∆︸︷︷︸n-times

→ R+ is called a n-D

grid-control if it is continuous and partially super-additive, i.e. for all (s1, t1) , . . . , (sn, tn) ∈ ∆and si < ui < ti we have

ω ([s1, t1]× . . .× [si, ui]× . . .× [sn, tn]) + ω ([s1, t1]× . . .× [ui, ti]× . . .× [sn, tn])

≤ ω ([s1, t1]× . . .× [si, ti]× . . .× [sn, tn])

for every i = 1, . . . , n. ω is called symmetric if

ω ([s1, t1]× . . .× [sn, tn]) = ω([sσ(1), tσ(1)

]× . . .×

[sσ(n), tσ(n)

])holds for every σ ∈ Sn.

The point of this definition is that |f (A)|p ≤ ω (A) for every rectangle A ⊂ [0, 1]n impliesthat Vp (f,R)p ≤ ω (R) for every rectangle R ⊂ [0, 1]n. Note that a 2D control is automaticallya 2D grid-control. The following immediate properties will be used in Section 2.3.2 with m =n = 2.

Lemma 2.2.3. (i) The restriction of a (m+ n)-dimensional grid-control to m arguments isa m-dimensional grid-control.

(ii) The product of a m- and a n-dimensional grid-control is a (m + n)-dimensional grid-control.

44

Multidimensional Young-integration and grid-controls

2.1 Iterated 2D-integralsIn the 1-dimensional case, the classical Young-theory allows to define iterated integrals of

functions with finite p-variation where p < 2. There, the superadditivity of (s, t) 7→ |·|pp−var;[s,t]played an essential role. We will see that Theorem 0.0.1 can be used to define and estimateiterated 2D-integrals. This will play an important role in Section 2.3 when we estimate theL2-norm of iterated integrals of Gaussian processes.

Lemma 2.2.4. Let f, g : [0, 1]2 → R be continuous where f has finite p-variation and g finitecontrolled q-variation with p−1 + q−1 > 1. Let (s, t) ∈ ∆ and assume that f (s, ·) = f (·, s) = 0.Define Φ: [s, t]2 → R by

Φ (u, v) =

∫[s,u]×[s,v]

f dg.

Then there is a constant C = C (p, q) such that

Vq−var

(Φ; [s, t]2

)≤ C (p, q)Vp−var

(f ; [s, t]2

)|g|q−var;[s,t]2 .

Proof. (i) Let ti < ti+1 and tj < tj+1. Then,

Φ

(ti, ti+1

tj , tj+1

)=

∫[ti,ti+1]×[tj ,tj+1]

f dg.

Now let ti < u < ti+1 and tj < v < tj+1. Then one has

f

(ti, utj , v

)= f (u, v)− f (ti, v)− f

(u, tj

)+ f

(ti, tj

).

Therefore,∣∣∣∣Φ(ti, ti+1

tj , tj+1

)∣∣∣∣ ≤∣∣∣∣∣∫

[ti,ti+1]×[tj ,tj+1]f

(ti, utj , v

)dg (u, v)

∣∣∣∣∣+

∣∣∣∣∣∫

[ti,ti+1]×[tj ,tj+1]f (ti, v) dg (u, v)

∣∣∣∣∣+

∣∣∣∣∣∫

[ti,ti+1]×[tj ,tj+1]f(u, tj

)dg (u, v)

∣∣∣∣∣+

∣∣∣∣∣∫

[ti,ti+1]×[tj ,tj+1]f(ti, tj

)dg (u, v)

∣∣∣∣∣For the first integral we use Young 2D-estimates to see that∣∣∣∣∣

∫[ti,ti+1]×[tj ,tj+1]

f

(ti, utj , v

)dg (u, v)

∣∣∣∣∣≤ c1 (p, q)Vp

(f, [ti, ti+1]×

[tj , tj+1

])Vq(g, [ti, ti+1]×

[tj , tj+1

])≤ c1 (p, q)Vp

(f, [s, t]2

)|g|q−var;[ti,ti+1]×[tj ,tj+1]

For the second, one has by a Young 1D-estimate∣∣∣∣∣∫

[ti,ti+1]×[tj ,tj+1]f (ti, v) dg (u, v)

∣∣∣∣∣ =

∣∣∣∣∣∫

[tj ,tj+1]f (ti, v) d (g (ti+1, v)− g (ti, v))

∣∣∣∣∣≤ c2 sup

u∈[s,t]|f (u, ·)|p−var;[s,t] |g|q−var;[ti,ti+1]×[tj ,tj+1] .

Similarly,∣∣∣∣∣∫

[ti,ti+1]×[tj ,tj+1]f(u, tj

)dg (u, v)

∣∣∣∣∣ ≤ c2 supv∈[s,t]

|f (·, v)|p−var;[s,t] |g|q−var;[ti,ti+1]×[tj ,tj+1] .

45


Finally,∣∣∣∣∣∫

[ti,ti+1]×[tj ,tj+1]f(ti, tj

)dg (u, v)

∣∣∣∣∣ =∣∣f (ti, tj)∣∣ ∣∣∣∣g(ti, ti+1

tj , tj+1

)∣∣∣∣ ≤ |f |∞;[s,t] |g|q−var;[ti,ti+1]×[tj ,tj+1] .

Putting all together, we get∣∣∣∣Φ(ti, ti+1

tj , tj+1

)∣∣∣∣q≤ c3

(Vp

(f, [s, t]2

)+ supu∈[s,t]

|f (u, ·)|p−var;[s,t] + supv∈[s,t]

|f (·, v)|p−var;[s,t] + |f |∞;[s,t]

)q× |g|q

q−var;[ti,ti+1]×[tj ,tj+1].

Take a partition D ⊂ [s, t] and u ∈ [s, t]. Then∑ti∈D|f (u, ti+1)− f (u, ti)|p =

∑ti∈D

∣∣∣∣f ( s, uti, ti+1

)∣∣∣∣p ≤ Vp (f, [s, t]2)pand hence

supu∈[s,t]

|f (u, ·)|p−var;[s,t] ≤ Vp(f, [s, t]2

).

The same way one obtains

supv∈[s,t]

|f (·, v)|p−var;[s,t] ≤ Vp(f, [s, t]2

).

Finally, for u, v ∈ [s, t],

|f (u, v)| =∣∣∣∣f (s, us, v

)∣∣∣∣ ≤ Vp (f, [s, t]2)and therefore |f |∞;[s,t] ≤ Vp

(f, [s, t]2

). Putting everything together, we end up with∣∣∣∣Φ(ti, ti+1

tj , tj+1

)∣∣∣∣q ≤ c4Vp

(f, [s, t]2

)q|g|q

q−var;[ti,ti+1]×[tj ,tj+1].

Hence for every partition D, D ⊂ [s, t] one gets, using superadditivity of |g|qq−var,∑ti∈D,tj∈D

∣∣∣∣Φ(ti, ti+1

tj , tj+1

)∣∣∣∣q ≤ c4Vp

(f, [s, t]2

)q ∑ti∈D,tj∈D

|g|qq−var;[ti,ti+1]×[tj ,tj+1]

≤ c4Vp

(f, [s, t]2

)q|g|q

q−var;[s,t]2 .

Passing to the supremum over all partitions shows the assertion.

This lemma allows us to define iterated 2D-integrals. Let f, g1, . . . , gn : [0, 1]2 → R. Aniterated 2D-integral is given by

∫∆1s,t×∆1

s′,t′f dg1 =

∫[s,t]×[s′,t′] f (u, v) dg1 (u, v) for n = 1 and

recursively defined by∫∆ns,t×∆n

s′,t′

f dg1 . . . dgn :=

∫[s,t]×[s′,t′]

(∫∆n−1s,u ×∆n−1

s′,v

f dg1 . . . dgn−1

)dgn (u, v)

for n ≥ 2.

46

The main estimates

Proposition 2.2.5. Let f, g1, g2, . . . : [0, 1]2 → R and p, q1, q2, . . . be real numbers such thatp−1 + q−1

1 > 1 and q−1i + q−1

i+1 > 1 for every i ≥ 1. Assume that f has finite p-variation andgi has finite qi-variation for i = 1, 2, . . . and that for (s, t) ∈ ∆ we have f (s, ·) = f (·, s) = 0.Then for every n ∈ N there is a constant C = C (p, q1, . . . , qn) such that∣∣∣∣∣

∫∆ns,t×∆n

s,t

f dg1 . . . dgn

∣∣∣∣∣ ≤ CVp (f, [s, t]2)Vq1 (g1, [s, t]2). . . Vqn

(gn, [s, t]

2).

Proof. Define Φ(n) (u, v) =∫

∆ns,u×∆n

s,vf dg1 . . . dgn. We will show a stronger result; namely that

for every n ∈ N and q′n > qn there is a constant C = C (p, q1, . . . , qn, q′n) such that

Vq′n

(Φ(n), [s, t]2

)≤ CVp

(f, [s, t]2

)Vq1

(g1, [s, t]

2). . . Vqn

(gn, [s, t]

2).

To do so, let q1, q2, . . .be a sequence of real numbers such that qj > qj and 1qj−1

+ 1qj> 1 for every

j = 1, 2, . . . where we set q0 = p. We make an induction over n. For n = 1, we have q1 > q1

and 1p + 1

q1> 1, hence from Theorem 0.0.1 we know that g1 has finite controlled q1-variation

and Lemma 2.2.4 gives us

Vq1

(Φ(1); [s, t]2

)≤ c1Vp

(f ; [s, t]2

)|g1|q1;[s,t]2 ≤ c2Vp

(f ; [s, t]2

)Vq1

(g1; [s, t]2

).

W.l.o.g, we may assume that q′1 > q1 > q1, otherwise we choose q1 smaller in the beginning.

From Vq′1

(Φ(1); [s, t]2

)≤ Vq1

(Φ(1); [s, t]2

)the assertion follows for n = 1. Now take n ∈ N.

Note that

Φ(n) (u, v) =

∫[s,u]×[s,v]

Φ(n−1) dgn

and clearly Φ(n−1) (s, ·) = Φ(n−1) (·, s) = 0. We can use Lemma 2.2.4 again to see that

Vqn

(Φ(n), [s, t]2

)≤ c3Vqn−1

(Φ(n−1); [s, t]2

)|gn|qn−var;[s,t]2

≤ c4Vqn−1

(Φ(n−1); [s, t]2

)Vqn

(gn; [s, t]2

).

Using our induction hypothesis shows the result for qn. By choosing qn smaller in the beginningif necessary, we may assume that q′n > qn and the assertion follows.

2.3 The main estimates

In the following section, (X,Y ) =(X1, Y 1, . . . , Xd, Y d

)will always denote a centred continuous

Gaussian process where(Xi, Y i

)and

(Xj , Y j

)are independent for i 6= j. We will also assume

that the ρ-variation of R(X,Y ) is finite for a ρ < 2 and controlled by a symmetric 2D-controlω (this in particular implies that the ρ-variation of RX , RY and RX−Y is controlled by ω, see[FV10b, Section 15.3.2]). Let γ > ρ such that 1

ρ + 1γ > 1. The aim of this section is to show

that for every n ∈ N there are constants C (n) such that5

∣∣Xns,t −Yn

s,t

∣∣L2((Rd)

⊗n) ≤ C (n) εω(

[s, t]2) 1

2γω(

[s, t]2)n−1

2ρfor every s < t (2.4)

where ε2 = V∞

(RX−Y , [s, t]

2)1−ρ/γ

(see Definition 2.3.1 below for the exact definition of V∞).

Equivalently, we might show (2.4) coordinate-wise, i.e. proving that the same estimate holds for

5We prefer to write it in this notation instead of writing ω([s, t]2

) 12γ

+n−12ρ to emphasize the different roles of

the two terms. The first term will play no particular role and just comes from interpolation whereas the secondone will be crucial when doing the induction step from lower to higher levels in Proposition 2.3.21.

47


|Xw −Yw|L2(R) for every word w formed by the alphabet A = 1, . . . , d. In some special cases,i.e. if a word w has a very simple structure, we can do this directly using multidimensional Youngintegration. This is done in Subsection 2.3.1. Subsection 2.3.2 shows (2.4) for n = 1, 2, 3, 4coordinate-wise, using the shuffle algebra structure for iterated integrals and multidimensionalYoung integration. In Subsection 2.3.3, we show (2.4) coordinate-free for all n > 4, using aninduction argument very similar to the one Lyon’s used for proving the Extension Theorem (cf.[Lyo98]).

We start with giving a 2-dimensional analogue for the one-dimensional interpolation inequal-ity.

Definition 2.3.1. If f : [0, 1]2 → B is a continuous function in a Banach space and (s, t) ×(u, v) ∈ ∆×∆ we set

V∞ (f, [s, t]× [u, v]) = supA⊂[s,t]×[u,v]

|f (A)| .

Lemma 2.3.2. For γ > ρ ≥ 1 we have the interpolation inequality

Vγ−var (f, [s, t]× [u, v]) ≤ V∞ (f, [s, t]× [u, v])1−ρ/γ Vρ−var (f, [s, t]× [u, v])ρ/γ

for all (s, t) , (u, v) ∈ ∆.

Proof. Exactly as 1D-interpolation, see [FV10b, Proposition 5.5].

3.1 Some special cases

If Z : [0, 1]→ R is a process with smooth sample paths, we will use the notation

Z(n)s,t =

∫∆ns,t

dZ . . . dZ

for s < t.

Lemma 2.3.3. Let X : [0, 1]→ R be a centred Gaussian process with continuous paths of finitevariation and assume that the ρ-variation of the covariance RX is controlled by a 2D-controlω. For fixed s < t, define

f (u, v) = E(X(n)s,uX(n)

s,v

).

Then there is a constant C = C (ρ, n) such that

Vρ

(f, [s, t]2

)≤ Cω

([s, t]2

)nρ.

Proof. Let ti < ti+1, tj < tj+1. Then

f

(ti, ti+1

tj , tj+1

)= E

((X

(n)s,ti+1

−X(n)s,ti

)(X

(n)

s,tj+1−X

(n)

s,tj

)).

We know that X(n) = (X)n

n! . From the identity

bn − an = (b− a)(an−1 + an−2b+ . . .+ . . . abn−2 + bn−1

)we deduce that

f

(ti, ti+1

tj , tj+1

)=

1

(n!)2

n−1∑k,l=0

E

(Xti,ti+1Xtj ,tj+1

(Xs,ti+1

)n−1−k(Xs,ti)

k(Xs,tj+1

)n−1−l (Xs,tj

)l).

48

The main estimates

We want to apply Wick’s formula now (cf. [Jan97, Theorem 1.28]). If

Z, Z ∈Xs,ti+1 , Xs,ti , Xs,tj+1

, Xs,tj

we know that ∣∣E (Xti,ti+1Z

)∣∣ρ ≤ ω ([ti, ti+1]× [s, t])∣∣∣E (Xti,ti+1Xtj ,tj+1

)∣∣∣ρ ≤ ω([ti, ti+1]×

[tj , tj+1

])∣∣∣E (ZZ)∣∣∣ρ ≤ ω(

[s, t]2)

and the same holds for Xtj ,tj+1. Now take two partitions D, D ∈ [0, 1]. Then, by Wick’s formula

and the estimates above,

∑ti∈D,tj∈D

∣∣∣∣f (ti, ti+1

tj , tj+1

)∣∣∣∣ρ ≤ c1 (ρ, n)ω(

[s, t]2)n−2 ∑

ti∈D,tj∈D

ω ([ti, ti+1]× [s, t])ω([tj , tj+1

]× [s, t]

)+c2 (ρ, n)ω

([s, t]2

)n−1 ∑ti∈D,tj∈D

ω([ti, ti+1]×

[tj , tj+1

])≤ c3ω

([s, t]2

)n.

Lemma 2.3.4. Let (X,Y ) be a centred Gaussian process in R2 with continuous paths of finitevariation. Assume that the ρ-variation of R(X,Y ) is controlled by a 2D-control ω for ρ < 2 andtake γ > ρ. Then for every n ∈ N there is a constant C = C (n) such that

∣∣∣X(n)s,t −Y

(n)s,t

∣∣∣L2≤ C (n) εω

([s, t]2

) 12γω(

[s, t]2)n−1

2ρ

for any s < t where ε2 = V∞

(RX−Y , [s, t]

2)1−ρ/γ

.

Proof. By induction. For n = 1 we simply have from Lemma 2.3.2

|Xs,t − Ys,t|2L2 = E [(Xs,t − Ys,t) (Xs,t − Ys,t)] ≤ Vγ−var(RX−Y , [s, t]

2)

≤ ε2Vρ−var

(RX−Y , [s, t]

2)ρ/γ

≤ ε2ω(

[s, t]2) 1γ

For n ∈ N we use the identity

X(n)s,t −Y

(n)s,t =

1

n

(Xs,tX

(n−1)s,t − Ys,tY(n−1)

s,t

)and hence∣∣∣X(n)

s,t −Y(n)s,t

∣∣∣L2≤ c1

(|Xs,t − Ys,t|L2

∣∣∣X(n−1)s,t

∣∣∣L2

+∣∣∣X(n−1)

s,t −Y(n−1)s,t

∣∣∣L2|Ys,t|L2

)≤ c2εω

([s, t]2

) 12γω(

[s, t]2)n−1

2ρ.

49


Assume that(Z1, Z2

)is a centred, continuous Gaussian process in R2 with smooth sample

paths and that both components are independent. Then (at least formally, cf. [FV10a]),∣∣∣∣∫ 1

0Z1

0,u dZ2u

∣∣∣∣2L2

= E

[(∫ 1

0Z1

0,u dZ2u

)2]

= E

[∫[0,1]2

Z10,uZ

10,v dZ

2 dZ2v

](2.5)

=

∫[0,1]2

E[Z1

0,uZ10,v

]dE[Z2uZ

2v

]=

∫[0,1]2

RZ1

(0 ·0 ·

)dRZ2 (2.6)

where the integrals in the second row are 2D Young-integrals (to make this rigorous, one usesthat the integrals are a.s. limits of Riemann sums and that a.s. convergence implies convergencein L1 in the (inhomogeneous) Wiener chaos). These kinds of computations together with ourestimates for 2D Young-integrals will be heavily used from now on.

Lemma 2.3.5. Let (X,Y ) =(X1, Y 1, . . . , Xd, Y d

)be a centred Gaussian process with contin-

uous paths of finite variation where(Xi, Y i

)and

(Xj , Y j

)are independent for i 6= j. Assume

that the ρ-variation of R(X,Y ) is controlled by a 2D-control ω for ρ < 2. Let w be a word of the

form w = i1 · · · in where i1, . . . , in ∈ 1, . . . , d are all distinct. Take γ > ρ such that 1ρ + 1

γ > 1.Then there is a constant C = C (ρ, γ, n) such that

∣∣Xws,t −Yw

s,t

∣∣L2 ≤ C (n) εω

([s, t]2

) 12γω(

[s, t]2)n−1

2ρ

for any s < t where ε2 = V∞

(RX−Y , [s, t]

2)1−ρ/γ

.

Proof. By the triangle inequality,

∣∣Xws,t −Yw

s,t

∣∣L2 =

∣∣∣∣∣∫

∆ns,t

dXi1 . . . dXin −∫

∆ns,t

dY i1 . . . dY in

∣∣∣∣∣L2

≤n∑k=1

∣∣∣∣∣∫

∆ns,t

dY i1 . . . dY ik−1 d(Xik − Y ik

)dXik+1 . . . dXin

∣∣∣∣∣L2

.

From independence, Proposition 2.2.5 and Lemma 2.3.2∣∣∣∣∣∫

∆ns,t

dY i1 . . . dY ik−1 d(Xik − Y ik

)dXik+1 . . . dXin

∣∣∣∣∣2

L2

=

∫∆ns,t×∆n

s,t

dRY i1 . . . dRY ik−1 dRXik−Y ik dRXik+1 . . . dRXin

≤ c1Vρ

(RY i1 , [s, t]

2). . . Vρ

(RY ik−1 , [s, t]

2)Vγ

(RXik−Y ik , [s, t]

2)

×Vρ(RXik+1 , [s, t]

2). . . Vρ

(RXin , [s, t]

2)

≤ c1Vγ

(RX−Y , [s, t]

2)ω(

[s, t]2)n−1

ρ ≤ c1ε2ω(

[s, t]2) 1γω(

[s, t]2)n−1

ρ.

The first inequality above is an immediate generalization of the calculations made in (2.5) and(2.6). Note that the respective random terms are not only pairwise but mutually independenthere since we are dealing with a Gaussian process (X,Y ). Interchanging the limits is allowedsince convergence in probability implies convergence in Lp, any p > 0, in the Wiener chaos.

3.2 Lower levels

50

The main estimates

n = 1, 2

Proposition 2.3.6. Let (X,Y ), ω, ρ and γ as in Lemma 2.3.5. Then there are constantsC (1) , C (2) which depend on ρ and γ such that

∣∣Xns,t −Yn

s,t

∣∣L2 ≤ C (n) εω

([s, t]2

) 12γω(

[s, t]2)n−1

2ρ

holds for n = 1, 2 and every (s, t) ∈ ∆ where ε2 = V∞

(RX−Y , [s, t]

2)1−ρ/γ

.

Proof. The coordinate-wise estimates are just special cases of Lemma 2.3.4 and Lemma 2.3.5.

n = 3

Proposition 2.3.7. Let (X,Y ), ω, ρ and γ as in Lemma 2.3.5. Then there is a constant C (3)which depends on ρ and γ such that

∣∣X3s,t −Y3

s,t

∣∣L2 ≤ C (3) εω

([s, t]2

) 12γω(

[s, t]2) 2

2ρ

holds for every (s, t) ∈ ∆ where ε2 = V∞

(RX−Y , [s, t]

2)1−ρ/γ

.

Proof. We have to show the estimate for Xi,j,k −Yi,j,k where i, j, k ∈ 1, . . . , d. From Propo-sition 2.1.8 and 2.1.5 it follows that it is enough to show the estimate for Xw −Yw where

w ∈ iii, ijk, iij : i, j, k ∈ 1, . . . , d distinct .

The cases w = iii and w = ijk are special cases of Lemma 2.3.4 and Lemma 2.3.5. The rest ofthis section is devoted to show the estimate for w = iij.

Lemma 2.3.8. Let (X,Y ) : [0, 1]→ R2 be a centred Gaussian process and consider

f (u, v) = E ((Xu − Yu)Xv) .

Assume that the ρ-variation of R(X,Y ) is controlled by a 2D-control ω where ρ ≥ 1. Let s < t

and consider a rectangle [σ, τ ]× [σ′, τ ′] ⊂ [s, t]2. Let γ > ρ. Then

Vγ−var(f, [σ, τ ]×

[σ′, τ ′

])≤ εω

([s, t]2

)1/2(1/ρ−1/γ)ω([σ, τ ]×

[σ′, τ ′

])1/γwhere ε2 = V∞

(RX−Y , [s, t]

2)1−ρ/γ

.

Proof. Let u < v and u′ < v′ ∈ [s, t]. Then∣∣E ((Xu,v − Yu,v)Xu′,v′)∣∣ ≤ |Xu,v − Yu,v|L2

∣∣Xu′,v′∣∣L2

≤ V∞

(RX−Y , [s, t]

2)1/2

Vρ−var

(R(X,Y ), [s, t]

2)1/2

and hence

supu<v,u′<v′

∣∣E ((Xu,v − Yu,v)Xu′,v′)∣∣ ≤ V∞ (RX−Y , [s, t]2)1/2

ω(

[s, t]2) 1

2ρ.

51


Now take a partition D of [σ, τ ] and a partition D of [σ′, τ ′]. Then∑ti∈D,tj∈D

∣∣∣E ((Xti,ti+1 − Yti,ti+1

)Xtj ,tj+1

)∣∣∣γ≤ sup

u<v,u′<v′

∣∣E ((Xu,v − Yu,v)Xu′,v′)∣∣γ−ρ ∑

ti∈D,tj∈D

∣∣∣E ((Xti,ti+1 − Yti,ti+1

)Xtj ,tj+1

)∣∣∣ρ≤ V∞

(RX−Y , [s, t]

2)1/2(γ−ρ)

ω(

[s, t]2)1/2(γ/ρ−1)

ω([σ, τ ]×

[σ′, τ ′

])and taking the supremum over all partitions shows the result.

Lemma 2.3.9. Let (X,Y ) : [0, 1] → R2 be a centred Gaussian process with continuous pathsof finite variation. Assume that the ρ-variation of R(X,Y ) is controlled by a 2D-control ω whereρ ≥ 1. Consider the function

g (u, v) = E[(

X(2)s,u −Y(2)

s,u

)(X(2)s,v −Y(2)

s,v

)].

Then for every γ > ρ there is a constant C = C (ρ, γ) such that

Vγ−var

(g, [s, t]2

)≤ Cε2ω

([s, t]2

)1/γ+1/ρ


(RX−Y , [s, t]

2)1−ρ/γ

.

Proof. Let u < v and u′ < v′. Then

g

(u, vu′, v′

)= E

[((X(2)s,v −X(2)

s,u

)−(Y(2)s,v −Y(2)

s,u

))((X

(2)s,v′ −X

(2)s,u′

)−(Y

(2)s,v′ −Y

(2)s,u′

))]=

1

22E[((

X2s,v −X2

s,u

)−(Y 2s,v − Y 2

s,u

)) ((X2s,v′ −X2

s,u′)−(Y 2s,v′ − Y 2

s,u′))]

.

Now, (X2s,v −X2

s,u

)−(Y 2s,v − Y 2

s,u

)= Xu,v (Xs,u +Xs,v)− Yu,v (Ys,u + Ys,v)

= Xu,v (Xs,u − Ys,u) + (Xu,v − Yu,v)Ys,u+Xu,v (Xs,v − Ys,v) + (Xu,v − Yu,v)Ys,v.

The same way one gets(X2s,v′ −X2

s,u′)−(Y 2s,v′ − Y 2

s,u′)

= Xu′,v′(Xs,u′ − Ys,u′

)+(Xu′,v′ − Yu′,v′

)Ys,u′

+Xu′,v′(Xs,v′ − Ys,v′

)+(Xu′,v′ − Yu′,v′

)Ys,v′ .

Now we expand the product of both sums and take expectation. For the first term we obtain,using the Wick formula and Lemma 2.3.8,∣∣E (Xu,v (Xs,u − Ys,u)Xu′,v′

(Xs,u′ − Ys,u′

))∣∣≤

∣∣E (Xu,vXu′,v′)E[(Xs,u − Ys,u)


)]∣∣+∣∣E [Xu,v


)]E[Xu′,v′ (Xs,u − Ys,u)

]∣∣+∣∣E [Xu′,v′


)]E [Xu,v (Xs,u − Ys,u)]

∣∣≤ Vρ−var

(R(X,Y ), [u, v]×

[u′, v′

])Vγ−var

(RX−Y , [s, t]

2)

+2Vγ−var(R(X,X−Y ), [u, v]× [s, t]

)Vγ−var

(R(X,X−Y ),

[u′, v′

]× [s, t]

)≤ ε2ω

([u, v]×

[u′, v′

])1/ρω(

[s, t]2)1/γ

+2ε2ω(

[s, t]2)1/ρ−1/γ

ω ([u, v]× [s, t])1/γ ω([u′, v′

]× [s, t]

)1/γ.

52

The main estimates

Now take two partitions D, D of [s, t]. With our calculations above,∑ti∈D,tj∈D

∣∣∣E (Xti,ti+1 (Xs,ti − Ys,ti)Xtj ,tj+1

(Xs,tj

− Ys,tj))∣∣∣γ

≤ c1ε2γω

([s, t]2

) ∑ti∈D,tj∈D

ω([ti, ti+1]×

[tj , tj+1

])γ/ρ+c2ε

2γω(

[s, t]2)γ/ρ−1 ∑

ti∈D,tj∈D

ω ([ti, ti+1]× [s, t])ω([tj , tj+1

]× [s, t]

)≤ c3ε

2γ

(ω(

[s, t]2)ω(

[s, t]2)γ/ρ

+ ω(

[s, t]2)γ/ρ−1

ω(

[s, t]2)2).

The other terms are treated exactly the same way. Taking the supremum over all partitionsshows the result.

The next corollary completes the proof of Proposition 2.3.7.

Corollary 2.3.10. Let (X,Y ), ω, ρ and γ as in Lemma 2.3.5. Then there is a constantC = C (ρ, γ) such that ∣∣∣Xi,i,j

s,t −Yi,i,js,t

∣∣∣L2≤ Cεω

([s, t]2

) 12γω(

[s, t]2) 2

2ρ

holds for every (s, t) ∈ ∆ and i 6= j where ε2 = V∞

(RX−Y , [s, t]

2)1−ρ/γ

.

Proof. From the triangle inequality,∣∣∣Xi,i,js,t −Yi,i,j

s,t

∣∣∣L2≤

∣∣∣∣∣∫

[s,t]

(Xi,is,u −Yi,i

s,u

)dY j

u

∣∣∣∣∣L2

+

∣∣∣∣∣∫

[s,t]Yi,is,u d

(Xj − Y j

)u

∣∣∣∣∣L2

.

For the first integral, we use independence to move the expectation inside the integral as seenin the proof of Lemma 2.3.5, then we use 2D Young integration and Lemma 2.3.9 to obtain thedesired estimate. The second integral is estimated in the same way using Lemma 2.3.3.

n = 4

Proposition 2.3.11. Let (X,Y ), ω, ρ and γ as in Lemma 2.3.5. Then there is a constantC (4) which depends on ρ and γ such that

∣∣X4s,t −Y4

s,t

∣∣L2 ≤ C (4) εω

([s, t]2

) 12γω(

[s, t]2) 3

2ρ


(RX−Y , [s, t]

2)1−ρ/γ

.

Proof. From Proposition 2.1.8 and 2.1.5 one sees that it is enough to show the estimate forXw −Yw where

w ∈ iiii, ijkl, iijj, iiij, iijk, jiik : i, j, k, l ∈ 1, . . . , d distinct .

The cases w = iiii and w = ijkl are special cases of Lemma 2.3.4 and Lemma 2.3.5. Hence itremains to show the estimate for

w ∈ iijj, iiij, iijk, jiik : i, j, k ∈ 1, . . . , d pairwise distinct .

This is the content of the remaining section.

53


Lemma 2.3.12. Let (X,Y ), ω, ρ and γ as in Lemma 2.3.5. Then there is a constant C =C (ρ, γ) such that ∣∣∣Xi,i,j,k

s,t −Yi,i,j,ks,t


([s, t]2

) 12γω(

[s, t]2) 3

2ρ

holds for every (s, t) ∈ ∆ where i, j, k are distinct and ε2 = V∞

(RX−Y , [s, t]

2)1−ρ/γ

.

Proof. From the triangle inequality,∣∣∣Xi,i,j,ks,t −Yi,i,j,k

s,t

∣∣∣L2

=

∣∣∣∣∣∫s<u<v<t

Xi,is,u dX

ju dX

kv −

∫s<u<v<t

Yi,is,u dY

ju dY

kv

∣∣∣∣∣L2

≤

∣∣∣∣∣∫s<u<v<t

(Xi,is,u −Yi,i

s,u

)dXj

u dXkv

∣∣∣∣∣L2

+

∣∣∣∣∣∫s<u<v<t

Yi,is,u d

(Xj − Y j

)udXk

v

∣∣∣∣∣L2

+

∣∣∣∣∣∫s<u<v<t

Yi,is,u dY

ju d(Xk − Y k

)v

∣∣∣∣∣L2

.

For the first integral, we use Proposition 2.2.5 and Lemma 2.3.9 to obtain∣∣∣∣∣∫s<u<v<t

(Xi,is,u −Yi,i

s,u

)dXj

u dXkv

∣∣∣∣∣2

L2

=

∫∆2s,t×∆2

s,t

E[(

Xi,is,· −Yi,i

s,·) (

Xi,is,· −Yi,i

s,·)]dRXj dRXk

≤ c1ε2ω(

[s, t]2)1/γ+1/ρ

ω(

[s, t]2)2/ρ

.

For the other two integrals we also use Proposition 2.2.5 together with Lemma 2.3.3 to obtainthe same estimate.

Lemma 2.3.13. Let (X,Y ) : [0, 1]→ R2 be a centred Gaussian process with continuous pathsof finite variation. Assume that the ρ-variation of R(X,Y ) is controlled by a 2D-control ω whereρ ≥ 1. Consider the function

g (u, v) = E[(

X(3)s,u −Y(3)

s,u

)(X(3)s,v −Y(3)

s,v

)].

Then for every γ > ρ there is a constant C = C (ρ, γ) such that

Vγ−var

(g, [s, t]2

)≤ Cε2ω

([s, t]2

)1/γ+2/ρ


(RX−Y , [s, t]

2)(1−ρ/γ)

.

Proof. Similar to the one of Lemma 2.3.9 applying again Wick’s formula.

Corollary 2.3.14. Let (X,Y ), ω, ρ and γ as in Lemma 2.3.5. Then there is a constantC = C (ρ, γ) such that ∣∣∣Xi,i,i,j

s,t −Yi,i,i,js,t


([s, t]2

) 12γω(

[s, t]2) 3

2ρ


(RX−Y , [s, t]

2)(1−ρ/γ)

.

54

The main estimates

Proof. The triangle inequality gives

∣∣∣Xi,i,i,js,t −Yi,i,i,j

s,t

∣∣∣L2

=

∣∣∣∣∣∫

[s,t]Xi,i,is,u dX

ju −

∫[s,t]

Yi,i,is,u dY

ju

∣∣∣∣∣≤

∣∣∣∣∣∫

[s,t]

(Xi,i,is,u −Yi,i,i

s,u

)dXj

u

∣∣∣∣∣L2

+

∣∣∣∣∣∫

[s,t]Yi,i,is,u d

(Xj − Y j

)u

∣∣∣∣∣L2

.

For the first integral, we move the expectation inside the integral, use 2D Young integrationand Lemma 2.3.13 to conclude the estimate. The second integral is estimated the same wayapplying Lemma 2.3.3.

It remains to show the estimates for Xw − Yw where w ∈ iijj, jiik. We need to be abit careful here for the following reason: It is clear that Xi,i,j

0,1 =∫

[0,1] Xi,iu dXj

u. One might

expect that also Xj,i,i0,1 =

∫[0,1]X

ju dX

i,iu holds, but this is not true in general. Indeed, just take

f (u) = g (u) = u. Then∫ 1

0f (u) d

(∫ u

0g (v) dg (v)

)=

1

2

∫ 1

0u d(u2)

=

∫ 1

0u2 du =

1

3

but ∫∆2

0,1

f (u) dg (u) dg (v) =

∫∆3

0,1

du1 du2 du3 =1

6.

One the other hand, if g is smooth, we can use Fubini to see that∫∆2

0,1

f (u) dg (u) dg (v) =

∫[0,1]2

f (u) g′ (u) g′ (v) 1u<v du dv

=1

2

∫[0,1]2

f (u) g′ (u) g′ (v) 1u<v du dv

+1

2

∫[0,1]2

f (v) g′ (v) g′ (u) 1v<u du dv

=1

2

∫[0,1]2

(f (u) 1u<v + f (v) 1v<u

)g′ (u) g′ (v) du dv

=1

2

∫[0,1]2

f (u ∧ v) g′ (u) g′ (v) du dv

=1

2

∫[0,1]2

f (u ∧ v) d (g (u) g (v))

where the last integral is a 2D Young integral. Hence we have seen that an iterated 1D-integralcan be transformed into a usual 2D-integral. We will use this trick for the remaining estimates.

Lemma 2.3.15. Let f : [0, 1]2 → R be a continuous function. Set

f (u1, u2, v1, v2) = f (u1 ∧ u2, v1 ∧ v2) .

(i) Let u1 < u1, u2 < u2, v1 < v1, v2 < v2 be all in [0, 1]. Then

f

u1, u1

u2, u2

v1, v1

v2, v2

= f

(u, uv, v

)

55


where we set

[u, u] =

[u1, u1] ∩ [u2, u2] if [u1, u1] ∩ [u2, u2] 6= ∅

[0, 0] if [u1, u1] ∩ [u2, u2] = ∅ .

[v, v] =

[v1, v1] ∩ [v2, v2] if [v1, v1] ∩ [v2, v2] 6= ∅

[0, 0] if [v1, v1] ∩ [v2, v2] = ∅

(ii) For s < t, σ < t and p ≥ 1 we have

Vp (f, [s, t]× [σ, τ ]) = Vp

(f , [s, t]2 × [σ, τ ]2

).

Proof. (i) By definition of the higher dimensional increments,

f

u1, u1

u2, u2

v1

v2

= f

u1

u2

v1

v2

− f

u1

u2

v1

v2

− f

u1

u2

v1

v2

+ f

u1

u2

v1

v2

= f (u1 ∧ u2, v1 ∧ v2)− f (u1 ∧ u2, v1 ∧ v2)

−f (u1 ∧ u2, v1 ∧ v2) + f (u1 ∧ u2, v1 ∧ v2) .

By a case distinction, one sees that this is equal to f (u, v1 ∧ v2)−f (u, v1 ∧ v2). One goeson with

f

u1, u1

u2, u2

v1, v1

v2, v2

= f

u1, u1

u2, u2

v1

v2

− f

u1, u1

u2, u2

v1

v2

− f

u1, u1

u2, u2

v1

v2

+ f

u1, u1

u2, u2

v1

v2

= h (v1 ∧ v2)− h (v1 ∧ v2)− h (v1 ∧ v2) + h (v1 ∧ v2)

= h (v)− h (v)

where h (·) = f (u, ·)− f (u, ·) .Hence

h (v)− h (v) = f (u, v)− f (u, v)− f (u, v) + f (u, v) = f

(u, uv, v

).

(ii) Let D be a partition of [s, t] and D a partition of [σ, τ ]. Then by 1,

∑ti∈D,t∈D

∣∣∣∣f ( ti, ti+1

tj , tj+1

)∣∣∣∣p =∑

ti∈D,t∈D

∣∣∣∣∣∣∣∣f

ti, ti+1

ti, ti+1

tj , tj+1

tj , tj+1

∣∣∣∣∣∣∣∣p

≤(Vp

(f , [s, t]2 × [σ, τ ]2

))p,

hence Vp (f, [s, t]× [σ, τ ]) ≤ Vp

(f , [s, t]2 × [σ, τ ]2

). Now let D1, D2 be partitions of [s, t]

and D1, D2 be partitions of [σ, τ ]. Set D = D1 ∪D2, D = D1 ∪ D2. Then D is a partitionof [s, t] and D a partition of [σ, τ ] (see Figure 1 below). By (1),

∑t1i1∈D1,t2i2

∈D2

t1j1∈D1,t2j2

∈D2

∣∣∣∣∣∣∣∣f

t1i1 , t1i1+1

t2i2 , t2i2+1

t1j1 , t1j1+1

t2j2 , t2j2+1

∣∣∣∣∣∣∣∣p

=∑

ti∈D,t∈D

∣∣∣∣f ( ti, ti+1

tj , tj+1

)∣∣∣∣p ≤ (Vp (f, [s, t]× [σ, τ ]))p

and we also get Vp

(f , [s, t]2 × [σ, τ ]2

)≤ Vp (f, [s, t]× [σ, τ ]).

56

The main estimates

Lemma 2.3.16. Let (X,Y ) : [0, 1]→ R2 be a centred Gaussian process with continuous pathsof finite variation and assume that ω is a symmetric control which controls the ρ-variation of

R(X,Y ) where ρ ≥ 1. Take (s, t) ∈ ∆, γ > ρ and set ε2 = V∞

(RX−Y , [s, t]

2)1−ρ/γ

.

(i) Set f (u1, u2, v1, v2) = E [Xu1Xu2Xv1Xv2 ]. Then there is a constant C1 = C1 (ρ) and asymmetric 4D grid-control ω1 which controls the ρ-variation of f and

Vρ

(f, [s, t]4

)≤ ω1

([s, t]4

)1/ρ= C1ω

([s, t]2

) 2ρ.

(ii) Set f (u1, u2, v1, v2) = E[X

(2)s,u1∧u2X

(2)s,v1∧v2

]. Then there is a constant C2 = C2 (ρ) such

that

Vρ

(f , [s, t]4

)≤ C2ω

([s, t]2

) 2ρ.

(iii) Set

g (u1, u2, v1, v2) = E [(Xu1Xu2 − Yu1Yu2) (Xv1Xv2 − Yv1Yv2)] .

Then there is a constant C3 = C3 (ρ, γ) and a symmetric 4D grid-control ω2 which controlsthe γ-variation of g and

Vγ

(g, [s, t]4

)≤ ω2

([s, t]4

)1/γ= C3ε

2ω(

[s, t]2)1/γ+1/ρ

.

(iv) Set

g (u1, u2, v1, v2) = E

[(X(2) −Y(2)

)s,u1∧u2

(X(2) −Y(2)

)s,v1∧v2

].

Then there is a constant C4 = C4 (ρ, γ) such that

Vγ

(g, [s, t]4

)≤ C4ε

2ω(

[s, t]2)1/γ+1/ρ

.

Proof. (i) Let u1 < u1, u2 < u2, v1 < v1, v2 < v2. By the Wick-formula,

|E [Xu1,u1Xu2,u2Xv1,v1Xv2,v2 ]|ρ

≤ 3ρ−1 |E [Xu1,u1Xu2,u2 ]E [Xv1,v1Xv2,v2 ]|ρ + 3ρ−1 |E [Xu1,u1Xv1,v1 ]E [Xu2,u2Xv2,v2 ]|ρ

+3ρ−1 |E [Xu1,u1Xv2,v2 ]E [Xu2,u2Xv1,v1 ]|ρ

≤ 3ρ−1ω ([u1, u1]× [u2, u2])ω ([v1, v1]× [v2, v2])

+3ρ−1ω ([u1, u1]× [v1, v1])ω ([u2, u2]× [v2, v2])

+3ρ−1ω ([u1, u1]× [v2, v2])ω ([u2, u2]× [v1, v1])

= : ω1 ([u1, u1]× [u2, u2]× [v1, v1]× [v2, v2]) .

It is easy to see that ω1 is a symmetric grid-control and that it fulfils the stated property.

(ii) A direct consequence of Lemma 2.3.3 and Lemma 2.3.15.

(iii) We have

Xu1Xu2 − Yu1Yu2 = (Xu1 − Yu1)Xu2 + Yu1 (Xu2 − Yu2) .

57


Hence for u1 < u1, u2 < u2, v1 < v1, v2 < v2,

f

u1, u1

u2, u2

v1, v1

v2, v2

= E[(X − Y )u1,u1 Xu2,u2 (X − Y )v1,v1 Xv2,v2

]

+E[Yu1,u1 (X − Y )u2,u2 (X − Y )v1,v1 Xv2,v2

]+E

[(X − Y )u1,u1 Xu2,u2Yv1,v1 (X − Y )v2,v2

]+E

[Yu1,u1 (X − Y )u2,u2 Yv1,v1 (X − Y )v2,v2

]For the first term we have, using Lemma 2.3.8,∣∣∣E [(X − Y )u1,u1 Xu2,u2 (X − Y )v1,v1 Xv2,v2

]∣∣∣γ≤ 3γ−1

∣∣∣E [(X − Y )u1,u1 Xu2,u2

]∣∣∣γ ∣∣∣E [(X − Y )v1,v1 Xv2,v2

]∣∣∣γ+3γ−1

∣∣∣E [(X − Y )u1,u1 (X − Y )v1,v1

]∣∣∣γ |E [Xu2,u2Xv2,v2 ]|γ

+3γ−1∣∣∣E [(X − Y )u1,u1 Xv2,v2

]∣∣∣γ ∣∣∣E [Xu2,u2 (X − Y )v1,v1

]∣∣∣γ≤ 3γ−1ε2γω

([s, t]2

) γρ−1ω ([u1, u1]× [u2, u2])ω ([v1, v1]× [v2, v2])

+3γ−1ε2γω ([u1, u1]× [v1, v1])ω ([u2, u2]× [v2, v2])γρ

+3γ−1ε2γω(

[s, t]2) γρ−1ω ([u1, u1]× [v2, v2])ω ([u2, u2]× [v1, v1])

≤ 3γ−1ε2γω(

[s, t]2) γρ−1

(ω ([u1, u1]× [u2, u2])ω ([v1, v1]× [v2, v2])

+ω ([u1, u1]× [v1, v1])ω ([u2, u2]× [v2, v2])

+ω ([u1, u1]× [v2, v2])ω ([u2, u2]× [v1, v1]))

= : ω ([u1, u1]× [u2, u2]× [v1, v1]× [v2, v2]) .

ω is a symmetric grid-control and fulfils the stated property. The other terms are treatedin the same way.

(iv) Follows from Lemma 2.3.9 and Lemma 2.3.15.

Corollary 2.3.17. Let (X,Y ), ω, ρ and γ as in Lemma 2.3.5. Then there is a constantC = C (ρ, γ) such that ∣∣∣Xi,i,j,j

s,t −Yi,i,j,js,t


([s, t]2

) 12γω(

[s, t]2) 3

2ρ


(RX−Y , [s, t]

2)1−ρ/γ

.

Proof. As seen before, we can use Fubini to obtain

Xi,i,j,js,t =

∫∆2s,t

Xi,is,u1 dX

ju1 dX

ju2 =

1

2

∫[s,t]2

Xi,is,u1∧u2 d

(Xju1X

ju2

)and hence ∣∣∣Xi,i,j,j

s,t −Yi,i,j,js,t

∣∣∣L2≤ 1

2

∣∣∣∣∣∫

[s,t]2

(Xi,is,u1∧u2 −Yi,i

s,u1∧u2

)d(Xju1X

ju2

)∣∣∣∣∣L2

+1

2

∣∣∣∣∣∫

[s,t]2Yi,is,u1∧u2 d

(Xju1X

ju2 − Y

ju1Y

ju2

)∣∣∣∣∣L2

.

58

The main estimates

We use a Young 4D-estimate and the estimates of Lemma 2.3.16 to see that∣∣∣∣∣∫

[s,t]2

(Xi,is,u1∧u2 −Yi,i

s,u1∧u2

)d(Xju1X

ju2

)∣∣∣∣∣2

L2

=

∫[s,t]4

E[(

Xi,is,u1∧u2 −Yi,i

s,u1∧u2

)(Xi,is,v1∧v2 −Yi,i

s,v1∧v2

)]dE[Xju1X

ju2X

jv1X

jv2

]≤ c1ε

2ω(

[s, t]2)1/γ

ω(

[s, t]2)3/ρ

.

The second term is estimated in the same way using again Lemma 2.3.16.

Lemma 2.3.18. Let f : [0, 1]2 → R and g : [0, 1]2 × [0, 1]2 → R be continuous where g issymmetric in the first and the last two variables. Let (s, t) ∈ ∆ and assume that f (s, ·) =f(·, s) = 0. Assume also that f has finite p-variation and that the q-variation of g is controlledby a symmetric 4D grid-control ω where 1

p + 1q > 1. Define

Ψ (u, v) =

∫[s,u]2×[s,v]2

f(u1 ∧ u2, v1 ∧ v2) dg (u1, u2; v1, v2)

Then there is a constant C = C (p, q) such that

Vq

(Ψ; [s, t]2

)≤ CVp

(f ; [s, t]2

)ω(

[s, t]4)1/q

.

Proof. Set

f (u1, u2, v1, v2) = f(u1 ∧ u2, v1 ∧ v2).

Let u < v and u′ < v′. Note that

1[s,v]2×[s,v′]2 − 1[s,u]2×[s,v′]2 − 1[s,v]2×[s,u′]2 + 1[s,u]2×[s,u′]2

= 1([s,v]2\[s,u]2)×[s,v′]2 − 1([s,v]2\[s,u]2)×[s,u′]2

= 1([s,v]2\[s,u]2)×([s,v′]2\[s,u′]2)

If we take out the square [s, u]2 of the larger square [s, v]2, what is left is the union of threeessentially disjoint squares. More precisely,

[s, v]2 \ [s, u]2 = [u, v]2 ∪ ([s, u]× [u, v]) ∪ ([u, v]× [s, u]) .

The same holds for u′ and v′. Hence,([s, v]2 \ [s, u]2

)×(

[s, v′]2 \ [s, u′]2)

=([u, v]2 ∪ ([s, u]× [u, v]) ∪ ([u, v]× [s, u])

)×([u′, v′]2 ∪

([s, u′]× [u′, v′]

)∪([u′, v′]× [s, u′]

))=

([u, v]2 × [u′, v′]2

)∪([u, v]2 × [s, u′]× [u′, v′]

)∪([u, v]2 × [u′, v′]× [s, u′]

)∪([s, u]× [u, v]× [u′, v′]2

)∪([s, u]× [u, v]× [s, u′]× [u′, v′]

)∪([s, u]× [u, v]× [u′, v′]× [s, u′]

)∪([u, v]× [s, u]× [u′, v′]2

)∪([u, v]× [s, u]× [s, u′]× [u′, v′]

)∪([u, v]× [s, u]× [u′, v′]× [s, u′]

)59


and all these are unions of essentially disjoint sets. Using continuity and the symmetry of fand g we have then

Ψ

(u, vu′, v′

)=

∫([s,v]2\[s,u]2)×([s,v′]2\[s,u′]2)

f dg

=

∫[u,v]2×[u′,v′]2

f dg + 2

∫[u,v]2×[s,u′]×[u′,v′]

f dg

+2

∫[s,u]×[u,v]×[u′,v′]2

f dg + 4

∫[s,u]×[u,v]×[s,u′]×[u′,v′]

f dg.

For the first integral we use Young 4D-estimates. Since f (s, ·, ·, ·) = . . . = f (·, ·, ·, s) = 0, wecan proceed as in the proof of Lemma 2.2.4 and use Lemma 2.3.15 to see that∣∣∣∣∣

∫[u,v]2×[u′,v′]2

f dg

∣∣∣∣∣ ≤ c1Vp

(f, [s, t]2

)Vq(g, [u, v]2 × [u′, v′]2

)≤ c1Vp

(f, [s, t]2

)ω([u, v]2 × [u′, v′]2

)1/qFor the second integral, we have∫

[u,v]2×[s,u′]×[u′,v′]f dg

=

∫[u,v]2×[s,u′]×[u′,v′]

f(u1 ∧ u2, v1 ∧ v2) dg (u1, u2; v1, v2)

=

∫[u,v]2×[s,u′]

f(u1 ∧ u2, v1) d[g(u1, u2; v1, v

′)− g (u1, u2; v1, u′)]

We now use a Young 3D-estimate to see that∣∣∣∣∣∫

[u,v]2×[s,u′]×[u′,v′]f dg

∣∣∣∣∣ ≤ c2Vp

(f (· ∧ ·, ·) , [s, t]3

)×Vq

(g(·, ·; ·, v′

)− g

(·, ·; ·, u′

), [u, v]2 ×

[s, u′

])As in Lemma 2.3.15, one can show that Vp

(f (· ∧ ·, ·) , [s, t]3

)= Vp

(f, [s, t]2

). For g, we have

Vq

(g(·, ·; ·, v′

)− g

(·, ·; ·, u′

), [u, v]2 ×

[s, u′

])≤ Vq

(g, [u, v]2 ×

[s, u′

]×[u′, v′

])≤ ω

([u, v]2 × [s, t]×

[u′, v′

])1/q.

Hence ∣∣∣∣∣∫

[u,v]2×[s,u′]×[u′,v′]f dg

∣∣∣∣∣ ≤ c2Vp

(f, [s, t]2

)ω(

[u, v]2 × [s, t]×[u′, v′

])1/q.

Similarly, using Young 3D and 2D estimates, we get∣∣∣∣∣∫

[s,u]×[u,v]×[u′,v′]2f dg

∣∣∣∣∣ ≤ c3Vp

(f, [s, t]2

)ω(

[s, t]× [u, v]×[u′, v′

]2)1/q

and ∣∣∣∣∣∫

[s,u]×[u,v]×[s,u′]×[u′,v′]f dg

∣∣∣∣∣ ≤ c4Vp

(f, [s, t]2

)ω([s, t]× [u, v]× [s, t]× [u′, v′]

)1/q.

60

The main estimates

Putting all together, using the symmetry of ω we have shown that∣∣∣∣Ψ( u, vu′, v′

)∣∣∣∣q ≤ c5Vp

(f, [s, t]2

)qω(

[u, v]× [u′, v′]× [s, t]2).

Since ω2 ([u, v]× [u′, v′]) := ω(

[u, v]× [u′, v′]× [s, t]2)

is a 2D grid-control this shows the claim.

We are now able to prove the remaining estimate.

Corollary 2.3.19. Let (X,Y ), ω, ρ and γ as in Lemma 2.3.5. Then there is a constantC = C (ρ, γ) such that ∣∣∣Xj,i,i,k

s,t −Yj,i,i,ks,t


([s, t]2

) 12γω(

[s, t]2) 3

2ρ

holds for every (s, t) ∈ ∆ and i, j, k pairwise distinct where ε2 = V∞

(RX−Y , [s, t]

2)1−ρ/γ

.

Proof. From ∫∆2s,w

Xjs,u1 dX

iu1 dX

iu2 =

1

2

∫[s,w]2

Xjs,u1∧u2 d

(Xiu1X

iu2

)we see that

Xj,i,i,ks,t =

1

2

∫ t

s

(∫[s,w]2

Xjs,u1∧u2 d

(Xiu1X

iu2

))dXk

w.

Hence ∣∣∣Xj,i,i,ks,t −Yj,i,i,k

s,t

∣∣∣L2

≤ 1

2

∣∣∣∣∫ t

sΨ1 (w) dXk

w

∣∣∣∣L2

+1

2

∣∣∣∣∫ t

sΨ2 (w) dXk

w

∣∣∣∣L2

+1

2

∣∣∣∣∫ t

sΨ3 (w) d

(Xk − Y k

)w

∣∣∣∣L2

where

Ψ1 (w) =

∫[s,w]2

(Xjs,u1∧u2 − Y

js,u1∧u2

)d(Xiu1X

iu2

)Ψ2 (w) =

∫[s,w]2

Y js,u1∧u2 d

(Xiu1X

iu2 − Y

iu1Y

iu2

)Ψ3 (w) =

∫[s,w]2

Y js,u1∧u2 d

(Y iu1Y

iu2

).

We start with the first integral. From independence and Young 2D-estimates,∣∣∣∣∫ t

sΨ1 (w) dXk

w

∣∣∣∣2L2

=

∫[s,t]2

E [Ψ1 (w1) Ψ1 (w2)] dE[Xkw1Xkw2

]≤ c1Vρ

(E [Ψ1 (·) Ψ1 (·)] , [s, t]2

)Vρ

(RXk [s, t]2

).

Now,

E [Ψ1 (w1) Ψ1 (w2)]

=

∫[s,w1]2×[s,w2]2

E[(Xjs,u1∧u2 − Y

js,u1∧u2

) (Xjs,v1∧v2 − Y

js,v1∧v2

)]dE[Xiu1X

iu2X

iv1X

iv2

].

61


In Lemma 2.3.16 we have seen that the ρ-variation of E[Xi·X

i·X

i·X

i·]

is controlled by a sym-metric grid-control ω1. Hence we can apply Lemma 2.3.18 to conclude that

Vρ

(E [Ψ1 (·) Ψ1 (·)] , [s, t]2

)≤ c2Vγ

(RX−Y ; [s, t]2

)ω1

([s, t]4

)1/ρ

≤ c3ε2ω(

[s, t]2)1/γ

ω(

[s, t]2)2/ρ

.

Clearly, Vρ

(RXk [s, t]2

)≤ ω

([s, t]2

)1/ρand therefore

∣∣∣∣∫ t

sΨ1 (w) dXk

w

∣∣∣∣2L2

≤ c4ε2ω(

[s, t]2)1/γ

ω(

[s, t]2)3/ρ

.

Now we come to the second integral. From independence,∣∣∣∣∫ t

sΨ2 (w) dXk

w

∣∣∣∣2L2

=

∫[s,t]2

E [Ψ2 (w1) Ψ2 (w2)] dE[Xkw1Xkw2

].

≤ c5Vγ

(E [Ψ2 (·) Ψ2 (·)] , [s, t]2

)Vρ

(RXk [s, t]2

).

Now

E [Ψ2 (w1) Ψ2 (w2)]

=

∫[s,w1]2×[s,w2]2

E[Y js,u1∧u2Y

js,v1∧v2

]dE[(Xiu1X

iu2 − Y

iu1Y

iu2

) (Xiv1X

iv2 − Y

iv1Y

iv2

)]= :

∫[s,w1]2×[s,w2]2

E[Y js,u1∧u2Y

js,v1∧v2

]dg (u1, u2, v1, v2) .

In Lemma 2.3.16 we have seen that the 4D γ-variation of g is controlled by a symmetric 4Dgrid-control ω2 where

ω2

([s, t]4

)1/γ= c6ε

2ω(

[s, t]2)1/ρ+1/γ

.

Hence

Vγ

(E [Ψ2 (·) Ψ2 (·)] , [s, t]2

)≤ c7Vρ

(RY j ; [s, t]2

)ω2

([s, t]4

)1/γ≤ c8ε

2ω(

[s, t]2)2/ρ+1/γ

.

This gives us ∣∣∣∣∫ t

sΨ2 (w) dXk

w

∣∣∣∣2L2

≤ c9ε2ω(

[s, t]2)1/γ

ω(

[s, t]2)3/ρ

.

For the third integral we see again that∣∣∣∣∫ t

sΨ3 (w) d

(Xk − Y k

)w

∣∣∣∣2L2

=

∫[s,t]2

E [Ψ3 (w1) Ψ3 (w2)] dE

[(Xk − Y k

)w1

(Xk − Y k

)w2

]≤ c10Vρ

(E [Ψ3 (·) Ψ3 (·)] , [s, t]2

)Vγ

(RX−Y , [s, t]

2).

From

E [Ψ3 (w1) Ψ3 (w2)] =

∫[s,w1]2×[s,w2]2

E[Y js,u1∧u2Y

js,v1∧v2

]dE[Y iu1Y

iu2Y

iv1Y

iv2

]we see that we can apply Lemma 2.3.18 to obtain

Vρ

(E [Ψ3 (·) Ψ3 (·)] , [s, t]2

)≤ c11Vρ

(RY j ; [s, t]2

)ω(

[s, t]2)2/ρ

≤ c11ω(

[s, t]2)3/ρ

.

62

The main estimates

Clearly, Vγ

(RX−Y , [s, t]

2)≤ ε2ω

([s, t]2

)1/γand hence

∣∣∣∣∫ t

sΨ3 (w) d

(Xk − Y k

)w

∣∣∣∣2L2

≤ c12ε2ω(

[s, t]2)1/γ

ω(

[s, t]2)3/ρ

which gives the claim.

Remark 2.3.20. Even though Proposition 2.3.6, 2.3.7 and 2.3.11 are only formulated for Gaus-sian processes with sample paths of finite variation, the estimate (2.4) is valid also for generalGaussian rough paths for n = 1, 2, 3, 4. Indeed, this follows from the fact that Gaussian roughpaths are just defined as L2 limits of smooth paths, cf. [FV10a].

3.3 Higher levelsOnce we have shown our desired estimates for the first four levels, we can use induction to

obtain also the higher levels. This is done in the next proposition.

Proposition 2.3.21. Let X and Y be Gaussian processes as in Theorem 2.0.1. Let ρ, γ befixed and ω be a control. Assume that there are constants C = C (n) such that

∣∣Xns,t

∣∣L2 ,∣∣Yn

s,t

∣∣L2 ≤ C (n)

ω (s, t)n2ρ

β(n2ρ

)!

holds for n = 1, . . . , [2ρ] and constants C = C (n) such that

∣∣Xns,t −Yn

s,t

∣∣L2 ≤ C (n) εω (s, t)

12γω (s, t)

n−12ρ

β(n−12ρ

)!

holds for n = 1, . . . , [2ρ] + 1 and every (s, t) ∈ ∆. Here, ε > 0 and β is a positive constant suchthat

β ≥ 4ρ

(1 + 2([2ρ]+1)/2ρ

(ζ

([2ρ] + 1

2ρ

)− 1

))where ζ is just the usual Riemann zeta function. Then for every n ∈ N there is a constantC = C (n) such that ∣∣Xn

s,t −Yns,t

∣∣L2 ≤ Cεω (s, t)

12γω (s, t)

n−12ρ

β(n−12ρ

)!

holds for every (s, t) ∈ ∆.

Proof. From Proposition 2.1.7 we know that for every n ∈ N there are constants C (n) suchthat ∣∣Xn

s,t

∣∣L2 ,∣∣Yn

s,t

∣∣L2 ≤ C

ω (s, t)n2ρ

β(n2ρ

)!

holds for all s < t. We will proof the assertion by induction over n. The induction basis isfulfiled by assumption. Suppose that the statement is true for k = 1, . . . , n where n ≥ [2ρ] + 1.We will show the statement for n+ 1. Let D = s = t0 < t1 < . . . < tj = t be any partition of[s, t]. Set

Xs,t : =(1,X1

s,t, . . . ,Xns,t, 0

)∈ Tn+1

(Rd),

XDs,t : = Xs,t1 ⊗ . . .⊗ Xtj−1,t

63


and the same for Y. We know that lim|D|→0 XDs,t = Sn+1 (X)s,t a.s. and the same holds

for Y (indeed, this is just the definition of the Lyons lift, cf. [Lyo98, Theorem 2.2.1]). Bymultiplicativity, πk

(XDs,t

)= Xk

s,t for k ≤ n. We will show that for any dissection D we have

∣∣πn+1

(XDs,t − YD

s,t

)∣∣L2 ≤ C (n+ 1) εω (s, t)

12γω (s, t)

n2ρ

β(n2ρ

)!.

We use the notation(XD)k

:= πk(XD). Assume that j ≥ 2. Let D′ be the partition of [s, t]

obtained by removing a point ti of the dissection D for which

ω (ti−1, ti+1) ≤

2ω(s,t)j−1 for j ≥ 3

ω (s, t) for j = 2

holds (Lemma 2.2.1 in [Lyo98] shows that there is indeed such a point). By the triangle in-equality,∣∣∣(XD −YD

)n+1∣∣∣L2≤∣∣∣∣(XD −XD′

)n+1−(YD −YD′

)n+1∣∣∣∣L2

+

∣∣∣∣(XD′ −YD′)n+1

∣∣∣∣L2

.

We estimate the first term on the right hand side. As seen in the proof of [Lyo98, Theorem

2.2.1],(XDs,t −XD′

s,t

)n+1=∑n

l=1 Xlti−1,tiX

n+1−lti,ti+1

. Set Rl = Yl−Xl. Then(XDs,t −XD′

s,t

)n+1−(YDs,t −YD′

s,t

)n+1

=n∑l=1

Xlti−1,tiX

n+1−lti,ti+1

−(Xlti−1,ti + Rl

ti−1,ti

)(Xn+1−lti,ti+1

+ Rn+1−lti,ti+1

)=

n∑l=1

−Xlti−1,tiR

n+1−lti,ti+1

−Rlti−1,tiY

n+1−lti,ti+1

.

By the triangle inequality, equivalence of Lq-norms in the Wiener Chaos, our moment estimatefor Xk and Yk and the induction hypothesis,∣∣∣∣(XD

s,t −XD′s,t


s,t

)n+1∣∣∣∣L2

≤ c1 (n+ 1)n∑l=1

∣∣∣Xlti−1,ti

∣∣∣L2

∣∣∣Rn+1−lti,ti+1

∣∣∣L2

+∣∣∣Rl

ti−1,ti

∣∣∣L2

∣∣∣Yn+1−lti,ti+1

∣∣∣L2

≤ c2 (n+ 1)n∑l=1

εω (ti, ti+1)12γω (ti−1, ti)

l2ρ

β(l

2ρ

)!

ω (ti, ti+1)n−l2ρ

β(n−l2ρ

)!

+εω (ti−1, ti)12γω (ti−1, ti)

l−12ρ

β(l−12ρ

)!

ω (ti, ti+1)n+1−l

2ρ

β(n+1−l

2ρ

)!

≤ 2c2εω (s, t)12γ

n∑l=0

ω (ti−1, ti)l2ρ

β(l

2ρ

)!

ω (ti, ti+1)n−l2ρ

β(n−l2ρ

)!

=4ρ

β2c2εω (s, t)

12γ

1

2ρ

n∑l=0

ω (ti−1, ti)l2ρ(

l2ρ

)!

ω (ti, ti+1)n−l2ρ(

n−l2ρ

)!

≤ 4ρc2εω (s, t)12γω (ti−1, ti+1)

n2ρ

β2(n2ρ

)!

64

The main estimates

where we used the neo-classical inequality (cf. [HH10]) and superadditivity of the controlfunction. Hence for j ≥ 3,∣∣∣∣(XD

s,t −XD′s,t


s,t

)n+1∣∣∣∣L2

≤ 4ρc2εω (s, t)12γω (ti−1, ti+1)

n2ρ

β2(n2ρ

)!

≤(

2

j − 1

) n2ρ

4ρc2εω (s, t)12γω (s, t)

n2ρ

β2(n2ρ

)!.

For j = 2 we get∣∣∣∣(XDs,t −XD′

s,t


s,t

)n+1∣∣∣∣L2

≤ 4ρc2εω (s, t)12γω (s, t)

n2ρ

β2(n2ρ

)!

but then D′ = s, t and therefore

∣∣∣∣(XD′s,t −YD′

s,t

)n+1∣∣∣∣L2

= 0. Hence by successively dropping

points we see that

∣∣∣(XDs,t −YD

s,t

)n+1∣∣∣L2≤

1 +

∞∑j=3

(2

j − 1

) n2ρ

4ρc2εω (s, t)12γω (s, t)

n2ρ

β2(n2ρ

)!

holds for all partitions D. Since n ≥ [2ρ] + 1,

∞∑j=3

(2

j − 1

) n2ρ

≤∞∑j=3

(2

j − 1

) [2ρ]+12ρ

≤ 2[2ρ]+1

2ρ

(ζ

([2ρ] + 1

2ρ

)− 1

)and thus

∣∣∣(XDs,t −YD

s,t

)n+1∣∣∣L2≤

4ρ

(1 + 2

[2ρ]+12ρ

(ζ(

[2ρ]+12ρ

)− 1))

βc2εω (s, t)

12γω (s, t)

n2ρ

β(n2ρ

)!.

By the choice of β, we get the uniform bound∣∣∣(XDs,t −YD

s,t

)n+1∣∣∣L2≤ c2εω (s, t)

12γω (s, t)

n2ρ

β(n2ρ

)!

which holds for all partitions D. Noting that a.s. convergence implies convergence in L2 in theWiener chaos, we obtain our claim by sending |D| → 0.

Corollary 2.3.22. Let (X,Y ), ω, ρ and γ as in Lemma 2.3.5. Then for all n ∈ N there areconstants C = C (ρ, γ, n) such that∣∣Xn

s,t −Yns,t

∣∣L2 ≤ Cεω

([s, t]2

) 12γ ω

([s, t]2

)n−12ρ


(RX−Y , [0, 1]2

)1−ρ/γ.

Proof. For n = 1, 2, 3, 4 this is the content of Proposition 2.3.6, 2.3.7 and 2.3.11. By makingthe constants larger if necessary, we also get

∣∣Xns,t −Yn

s,t

∣∣L2 ≤ c (n) εω

([s, t]2

) 12γω([s, t]2

)n−12ρ

β(n−12ρ

)!

65


with β chosen as in Proposition 2.3.21. We have already seen that

∣∣Xns,t

∣∣L2 ,∣∣Yn

s,t

∣∣L2 ≤ c (n)

ω([s, t]2

) n2ρ

β(n2ρ

)!

holds for constants c (n) where n = 1, 2, 3. Since ρ < 2, we have [2ρ] + 1 ≤ 4. From Proposition2.3.21 we can conclude that

∣∣Xns,t −Yn

s,t

∣∣L2 ≤ c (n) εω

([s, t]2

) 12γω([s, t]2

)n−12ρ

β(n−12ρ

)!

holds for every n ∈ N and constants c (n). Setting C (n) = c(n)

β(n−12ρ

)!

gives our claim.

2.4 Main result

Assume that X is a Gaussian process as in Theorem 2.0.1 with paths of finite p-variation.Consider a sequence (Λk)k∈N of continuous operators

Λk : Cp−var ([0, 1] ,R)→ C1−var ([0, 1] ,R) .

If x =(x1, . . . , xd

)∈ Cp−var

([0, 1] ,Rd

), we will write Λk (x) =

(Λk(x1), . . . ,Λk

(xd))

. Assumethat Λk fulfils the following conditions:

(i) Λk (x)→ x in the |·|∞-norm if k →∞ for every x ∈ Cp−var([0, 1] ,Rd

).

(ii) If RX has finite controlled ρ-variation, then, for some C = C (ρ),

supk,l∈N

∣∣R(Λk(X),Λl(X))

∣∣ρ−var;[0,1]2

≤ C |RX |ρ−var;[0,1]2 .

Our main result is the following:

Theorem 2.4.1. Let X be a Gaussian process as in Theorem 2.0.1 for ρ < 2 and K ≥Vρ

(RX , [0, 1]2

). Then there is an enhanced Gaussian process X with sample paths in

C0,p−var ([0, 1] , G[p](Rd))

w.r.t. (Λk)k∈N where p ∈ (2ρ, 4), i.e.∣∣ρp−var (S[p] (Λk (X)) ,X)∣∣Lr→ 0

for k → ∞ and every r ≥ 1. Moreover, choose γ such that γ > ρ and 1γ + 1

ρ > 1. Then forq > 2γ and every N ∈ N there is a constant C = C (q, ρ, γ,K,N) such that

|ρq−var (SN (Λk (X)) , SN (X))|Lr ≤ CrN/2 sup

0≤t≤1|Λk (X)t −Xt|

1− ργ

L2(Rd)

holds for every k ∈ N.

Proof. The first statement is a fundamental result about Gaussian rough paths, see [FV10b,Theorem 15.33]. For the second, take δ > 0 and set

γ′ = (1 + δ) γ and ρ′ = (1 + δ) ρ.

By choosing δ smaller if necessary we can assume that 1ρ′ + 1

γ′ > 1 and q > 2γ′. Set

ωk,l (A) =∣∣R(Λk(X),Λl(X))

∣∣ρ′ρ′−var;A

66

Main result

for a rectangle A ⊂ [0, 1]2 and

εk,l = V∞

(R(Λk(X)−Λl(X)), [0, 1]2

) 12− ρ′

2γ′= V∞

(R(Λk(X)−Λl(X)), [0, 1]2

) 12− ρ

2γ.

From Theorem 0.0.1 we know that ωk,l is a 2D control function which controls the ρ′-variationof R(Λk(X),Λl(X)). From Corollary 2.3.22 we can conclude that there is a constant c1 such that∣∣∣πn (SN (Λk (X))s,t − SN (Λl (X))s,t

)∣∣∣L2≤ c1εk,lωk,l

([s, t]2

) 12γ′ωk,l

([s, t]2

)n−12ρ′

holds for every n = 1, . . . , N , (s, t) ∈ ∆ and k, l ∈ N. Now,

ωk,l

([s, t]2

)n−12ρ′

=

ωk,l

([s, t]2

)ωk,l

([0, 1]2

)

n−12ρ′

ωk,l

([0, 1]2

)n−12ρ′

≤ ωk,l

([s, t]2

)n−12γ′

ωk,l

([0, 1]2

)n−12ρ′ −

n−12γ′

.

From Theorem 0.0.1 and our assumptions on the Λk we know that

ωk,l

([0, 1]2

)1/ρ′

≤ c2 |RX |ρ′−var;[0,1]2 ≤ c3Vρ

(RX , [0, 1]2

)≤ c4

(ρ, ρ′,K

).

holds uniformly over all k, l. Hence∣∣∣πn (SN (Λk (X))s,t − SN (Λl (X))s,t

)∣∣∣L2≤ c5εk,lωk,l

([s, t]2

) n2γ′.

Proposition 2.1.7 shows with the same argument that∣∣∣πn (SN (Λk (X))s,t

)∣∣∣L2≤ c6ωk,l

([s, t]2

) n2ρ′ ≤ c7ωk,l

([s, t]2

) n2γ′

for every k ∈ N and the same holds for SN (Λl (X))s,t. From [FV10b, Proposition 15.24] we canconclude that there is a constant c8 such that

|ρq−var (SN (Λk (X)) , SN (Λl (X)))|Lr ≤ c8rN/2εk,l

holds for all k, l ∈ N. In particular, we have shown that (SN (Λk (X)))k∈N is a Cauchy sequencein Lr and it is clear that the limit is given by the Lyons lift SN (X) of the enhanced Gaussianprocess X. Now fix k ∈ N. For every l ∈ N,

|ρq−var (SN (Λk (X)) , SN (X))|Lr ≤ |ρq−var (SN (Λk (X)) , SN (Λl (X)))|Lr+ |ρq−var (SN (Λl (X)) , SN (X))|Lr

≤ c8rN/2εk,l + |ρq−var (SN (Λl (X)) , SN (X))|Lr .

It is easy to see that

εk,l → V∞

(R(Λk(X)−X), [0, 1]2

) 12− ρ

2γfor l→∞

and since|ρq−var (SN (Λl (X)) , SN (X))|Lr → 0 for l→∞

we can conclude that

|ρq−var (SN (Λk (X)) , SN (X))|Lr ≤ c8rN/2V∞

(R(Λk(X)−X), [0, 1]2

) 12− ρ

2γ

67


holds for every k ∈ N. Finally, we have for [σ, τ ]× [σ′, τ ′] ⊂ [0, 1]2∣∣∣∣R(Λk(X)−X)

(σ, τσ′, τ ′

)∣∣∣∣Rd×d

≤ 4 sup0≤s<t≤1

∣∣R(Λk(X)−X) (s, t)∣∣Rd×d

and hence

V∞

(R(Λk(X)−X), [0, 1]2

)≤ 4 sup

0≤s<t≤1

∣∣R(Λk(X)−X) (s, t)∣∣Rd×d .

Furthermore, for any s < t,∣∣R(Λk(X)−X) (s, t)∣∣Rd×d ≤ |Λk (X)s −Xs|L2(Rd) |Λk (X)t −Xt|L2(Rd) ≤ sup

0≤t≤1|Λk (X)t −Xt|2L2(Rd)

and therefore

V∞

(R(Λk(X)−X), [0, 1]2

) 12− ρ

2γ ≤ c9 sup0≤t≤1

|Λk (X)t −Xt|1− ρ

γ

L2(Rd)

which shows the result.

The next Theorem gives pathwise convergence rates for the Wong-Zakai error for suitableapproximations of the driving signal.

Theorem 2.4.2. Let X be as in Theorem 2.0.1 for ρ < 2, K ≥ Vρ

(RX , [0, 1]2

)and X(k) =

Λk (X). Consider the SDEs

dYt = V (Yt) dXt, Y0 ∈ Rn (2.7)

dY(k)t = V (Y

(k)t ) dX

(k)t , Y

(k)0 = Y0 ∈ Rn (2.8)

where |V |Lipθ ≤ ν < ∞ for a θ > 2ρ. Assume that there is a constant C1 and a sequence(εk)k∈N ⊂

⋃r≥1

lr such that

sup0≤t≤1


∣∣∣2L2≤ C1ε

1/ρk for all k ∈ N.

Choose η, q such that

0 ≤ η < min

1

ρ− 1

2,

1

2ρ− 1

θ

and q ∈

(2ρ

1− 2ρη, θ

).

Then both SDEs (2.7) and (2.8) have unique solutions Y and Y (k) and there is a finite randomvariable C and a null set M such that∣∣∣Y (k) (ω)− Y (ω)

∣∣∣∞;[0,1]

≤∣∣∣Y (k) (ω)− Y (ω)

∣∣∣q−var;[0,1]

≤ C (ω) εηk (2.9)

holds for all k ∈ N and ω ∈ Ω \M . The random variable C depends on ρ, q, η, ν, θ,K,C1, thesequence (εk)k∈N and the driving process X but not on the equation itself. The same holds forthe set M .

Remark 2.4.3. Note that this means that we have universal rates, i.e. the set M and the ran-dom variable C are valid for all starting points (and also vector fields subject to a uniform Lipθ-bound). In particular, our convergence rates apply to solutions viewed as C l-diffeomorphismswhere l = [θ − q], cf. [FV10b, Theorem 11.12] and [FR11].

68

Main result

Proof of Theorem 2.4.2. Note that γ > ρ and 1ρ + 1

γ > 1 is equivalent to 0 < 12ρ −

12γ <

1ρ −

12 .

Hence there is a γ0 > ρ such that η = 12ρ −

12γ0

and 1ρ + 1

γ0> 1. Furthermore, 2γ0 = 2ρ

1−2ρη < q.

Choose γ1 > γ0 such that still 2γ1 < q and η < 12ρ −

12γ1

< 1ρ −

12 , hence 1

ρ + 1γ1> 1 hold. Set

α := 12ρ −

12γ1− η > 0. From Theorem 2.4.1 we know that for every r ≥ 1 and N ∈ N there is a

constant c1 such that∣∣∣ρq−var (SN (X(k)), SN (X))∣∣∣Lr≤ c1r

N/2 sup0≤t≤1


∣∣∣1− ργL2≤ c2r

N/2ε12ρ− 1

2γ

k

holds for every k ∈ N. Hence∣∣∣∣∣ρq−var(SN (X(k)), SN (X)

)εηk

∣∣∣∣∣Lr

≤ c2rN/2εαk

for every k ∈ N. From the Markov inequality, for any δ > 0,

∞∑k=1

P

[ρq−var

(SN (X(k)), SN (X)

)εηk

≥ δ

]≤ 1

δr

∞∑k=1

∣∣∣∣∣ρq−var(SN (X(k)), SN (X)

)εηk

∣∣∣∣∣r

Lr

≤ c3

∞∑k=1

εαrk

By assumption, we can choose r large enough such that the series converges. With Borel-Cantelliwe can conclude that

ρq−var(SN (X(k)), SN (X)

)εηk

→ 0

outside a null set M for k →∞. We set

C2 := supk∈N

ρq−var(SN (X(k)), SN (X)

)εηk

<∞ a.s.

Since C2 is the supremum of F-measurable random variables it is itself F-measurable. Now setN = [q] which turns ρq−var into a rough path metric. Note that since θ > 2ρ, (2.7) and (2.8)have indeed unique solutions Y and Y (k). We substitute the driver X by SN (X) resp. X(k) bySN (X(k)) in the above equations, now considered as RDEs in the q-rough paths space. Sinceθ > q, both (RDE-) equations have again unique solutions and it is clear that they coincidewith Y and Y (k). From

ρq−var

(SN (X(k)),1

)≤ ρq−var

(SN (X(k)), SN (X)

)+ρq−var (SN (X) ,1) ≤ C1+ρq−var (SN (X) ,1)

we see that for every ω ∈ Ω\M the SN (X(k) (ω)) are uniformly bounded for all k in the topologygiven by the metric ρq−var. Thus we can apply local Lipschitz-continuity of the Ito-Lyons map(see [FV10b, Theorem 10.26]) to see that there is a random variable C3 such that∣∣∣Y (k) − Y

∣∣∣q−var;[0,1]

≤ C3ρq−var

(SN (X(k)), SN (X)

)≤ C3 · C2ε

ηk

holds for every k ∈ N outside M . Finally,∣∣∣Y (k)t − Yt

∣∣∣ =∣∣∣Y (k)

0,t − Y0,t

∣∣∣ ≤ ∣∣∣Y (k) − Y∣∣∣q−var;[0,t]

≤∣∣∣Y (k) − Y

∣∣∣q−var;[0,1]

is true for all t ∈ [0, 1] and the claim follows.

69


4.1 Mollifier approximationsLet φ be a mollifier function with support [−1, 1], i.e. φ ∈ C∞0 ([−1, 1]) is positive and

|φ|L1 = 1. If x : [0, 1] → R is a continuous path, we denote by x : R → R its continuousextension to the whole real line, i.e.

xu =

x0 for x ∈ (−∞, 0]xu for x ∈ [0, 1]x1 for x ∈ [1,∞)

For ε > 0 set

φε (u) : =1

εφ (u/ε) and

xεt : =

∫Rφε (t− u) xu du.

Let (εk)k∈N be a sequence of real numbers such that εk → 0 for k →∞. Define

Λk (x) := xεk .

In [FV10b], Chapter 15.2.3 it is shown that the sequence (Λk)k∈N fulfils the conditions ofTheorem 2.4.1.

Corollary 2.4.4. Let X be as in Theorem 2.0.1 and assume that there is a constant C such that

Vρ

(RX ; [s, t]2

)≤ C |t− s|1/ρ holds for all s < t. Choose (εk)k∈N ∈

⋃r≥1

lr and set X(k) = Xεk .

Then the solutions Y (k) of the SDE (2.8) converge pathwise to the solution Y of (2.7) in thesense of (2.9) with rate O

(εηk)

where η is chosen as in Theorem 2.4.2.

Proof. It suffices to note that for every ε > 0, Z ∈X1, . . . , Xd

and t ∈ [0, 1] we have

E[|Zεt − Zt|

2]

= E

[(∫Rφε (t− u)

(Zu − Zt

)du

)2]

= E

(∫[t−ε,t+ε]

φε (t− u)(Zu − Zt

)du

)2

= E

[∫[t−ε,t+ε]2

φε (t− u)φε (t− v)(Zu − Zt

) (Zv − Zt

)du dv

]

=

∫[t−ε,t+ε]2

φε (t− u)φε (t− v)E[(Zu − Zt

) (Zv − Zt

)]du dv

≤ supt∈[0,1]|h1|,|h2|≤ε

∣∣E [(Zt+h1 − Zt) (Zt+h2 − Zt)]∣∣≤ sup

t∈[0,1]|h|≤ε

E[(Zt+h − Zt

)2] ≤ c1ε1/ρ

from which follows that sup0≤t≤1 |Xεkt −Xt|2L2 ≤ c1ε

1/ρk . We conclude with Theorem 2.4.2.

4.2 Piecewise linear approximationsIf D = 0 = t0 < t1 < . . . < t#D−1 = 1 is a partition of [0, 1] and x : [0, 1] → R a

continuous path, we denote by xD the piecewise linear approximation of x at the points of D,i.e. xD coincides with x at the points ti and if ti ≤ t < ti+1 we have

xDti+1− xDt

ti+1 − t=xti+1 − xtiti+1 − ti

.

70

Main result

Let (Dk)k∈N be a sequence of partitions of [0, 1] such that |Dk| := maxti∈Dk |ti+1 − ti| → 0for k →∞. If x : [0, 1]→ R is continuous, we define

Λk (x) := xDk .

In [FV10b, Chapter 15.2.3] it is shown that (Λk)k∈N fulfils the conditions of Theorem 2.4.1. IfRX is the covariance of a Gaussian process, we set

|D|RX ,ρ =

(maxti∈D

Vρ

(RX ; [ti, ti+1]2

))ρ.

Corollary 2.4.5. Let X be as in Theorem 2.0.1. Choose a sequence of partitions (Dk)k∈N of

the interval [0, 1] such that(|Dk|RX ,ρ

)k∈N∈⋃r≥1

lr and set X(k) = XDk . Then the solutions

Y (k) of the SDE (2.8) converge pathwise to the solution Y of (2.7) in the sense of (2.9) with

rate O(εηk)

where (εk)k∈N =(|Dk|RX ,ρ

)k∈N

and η is chosen as in Theorem 2.4.2.

Proof. Let D be any partition of [0, 1] and t ∈ [ti, ti+1] where ti, ti+1 ∈ D. Take Z ∈X1, . . . , Xd

. Then

ZDt − Zt = Zti,ti+1

t− titi+1 − ti

− Zti,t.

Therefore ∣∣ZDt − Zt∣∣L2 ≤∣∣Zti,ti+1

∣∣L2 + |Zti,t|L2 ≤ 2Vρ

(RX ; [ti, ti+1]2

)1/2≤ 2 |D|

12ρ

RX ,ρ.

We conclude with Theorem 2.4.2.

Example 2.4.6 Let X = BH be the fractional Brownian motion with Hurst parameterH ∈ (1/4, 1/2]. Set ρ = 1

2H < 2. Then one can show that RX has finite ρ-variation and

Vρ

(RX ; [s, t]2

)≤ c (H) |t− s|1/ρ for all (s, t) ∈ ∆ (see [FV11], Example 1). Assume that the

vector fields in (2.7) are sufficiently smooth by which we mean that 1/ρ− 1/2 ≤ 1/ (2ρ)− 1/θ,i.e.

θ ≥ 2ρ

ρ− 1=

1

1/2−H.

Let (Dk)k∈N be the sequence of uniform partitions. By Corollary 2.4.5, for every η < 2H − 1/2there is a random variable C such that∣∣∣Y (k) − Y

∣∣∣∞≤ C

(1

k

)ηa.s.

hence we have a Wong-Zakai convergence rate arbitrary close to 2H − 1/2. In particular, forthe Brownian motion, we obtain a rate close to 1/2, see also [GS06] and [FR11]. For H → 1/4,the convergence rate tends to 0 which reflects the fact that the Levy area indeed diverges forH = 1/4, see [CQ02].

4.3 The simplified step-N Euler scheme

Consider again the SDE

dYt = V (Yt) dXt, Y0 ∈ Rn

interpreted as a pathwise RDE driven by the lift X of a Gaussian process X which fulfils theconditions of Theorem 2.0.1. Let D be a partition of [0, 1]. We recall the simplified step-N

71


Euler scheme from the introduction:

Y sEulerN ;D0 = Y0

Y sEulerN ;Dtj+1

= Y sEulerN ;Dtj

+ Vi

(Y sEulerN ;Dtj

)Xitj ,tj+1

+1

2Vi1Vi2

(Y sEulerN ;Dtj

)Xi1tj ,tj+1

Xi2tj ,tj+1

+ . . .+1

N !Vi1 . . .V iN−1ViN

(Y sEulerN ;Dtj

)Xi1tj ,tj+1

. . . XiNtj ,tj+1

where tj ∈ D. In this section, we will investigate the convergence rate of this scheme. Forsimplicity, we will assume that

Vρ

(RX ; [s, t]2

)= O

(|t− s|1/ρ

)which can always be achieved at the price of a deterministic time-change based on

[0, 1] 3 t 7→Vρ

(RX ; [0, t]2

)ρVρ

(RX ; [0, 1]2

)ρ ∈ [0, 1] .

Set Dk =ik : i = 0, . . . , k

.

Corollary 2.4.7. Let p > 2ρ and assume that |V |Lipθ < ∞ for θ > p. Choose η and N suchthat

η < min

1

ρ− 1

2,

1

2ρ− 1

θ

and N ≤ [θ] .

Then there are random variables C1 and C2 such that

maxtj∈Dk

∣∣∣Ytj − Y sEulerN ;Dktj

∣∣∣ ≤ C1

(1

k

)η+ C2

(1

k

)N+1p−1

a.s. for all k ∈ N.

Proof. Recall the step-N Euler scheme from the introduction (or cf. [FV10b, Chapter 10]). Set

X(k) = XDk and let Y (k) be the solution of the SDE (2.8). Then Y sEulerN ;Dktj

=(Y (k)

)EulerN ;Dktj

for every tj ∈ Dk and therefore, using the triangle inequality,

maxtj∈Dk

∣∣∣Ytj − Y sEulerN ;Dktj

∣∣∣ ≤ supt∈[0,1]

∣∣∣Yt − Y (k)t

∣∣∣+ maxtj∈Dk

∣∣∣∣Y (k)tj−(Y (k)

)EulerN ;Dk

tj

∣∣∣∣ .By the choice of Dk we have |Dk|RX ,ρ = O

(k−1

). Applying Corollary 2.4.5 we obtain for the

first term∣∣Y − Y (k)

∣∣∞ = O (k−η). Refering to [FV10b, Theorem 10.30] we see that the second

term is of order O

(k−(N+1p−1))

.

Remark 2.4.8. Assume that the vector fields are sufficiently smooth, i.e. θ ≥ 2ρρ−1 . Then we

obtain an error of O(k−(2/p−1/2)

)+O

(k−(N+1p−1))

, any p > 2ρ. That means that in the case

ρ = 1, the step-2 scheme (i.e. the simplified Milstein scheme) gives an optimal convergence rateof (almost) 1/2. For ρ ∈ (1, 2), the step-3 scheme gives an optimal rate of (almost) 1/ρ− 1/2.In particular, we see that using higher order schemes does not improve the convergence ratesince in that case, the Wong-Zakai error persists. In the fractional Brownian motion case, thesimplified Milstein scheme gives an optimal convergence rate of (almost) 1/2 for the Brownianmotion and for H ∈ (1/4, 1/2) the step-3 scheme gives an optimal rate of (almost) 2H − 1/2.This answers a conjecture stated in [DNT12].

72

3

Integrability of (non-)linear roughdifferential equations and integrals

Integrability properties of linear rough differential equations (RDEs), and related topics, drivenby Brownian and then a Gaussian rough path (GRP), a random rough path X = X (ω), havebeen a serious difficulty in a variety of recent applications of rough path theory. To wit, for solu-tions of linear RDEs one has the typical - and as such sharp - estimateO (exp ((const)× ωx (0, T )))where ωx (0, T ) = ‖x‖pp-var;[0,T ] denotes some (homogeneous) p-variation norm (raised to power

p). In a Gaussian rough path setting, ‖X‖p-var;[0,T ] enjoys Gaussian integrability but as soon asp > 2 (the ”interesting” case, which covers Brownian and rougher situations) one has lost allcontrol over moments of such (random) linear RDE solutions. In a recent work, Cass, Litterer,Lyons [CLL] have overcome a similar problem, the integrability of the Jacobian of GaussianRDE flow, as needed in non-Markovian Hormander theory [HP].

With these (and some other, cf. below) problems in mind, we revisit the work of Cass,Litterer, Lyons and propose (what we believe to be) a particularly user-friendly formulation.We avoid the concept of ”localized p-variation”, as introduced in [CLL], and work throughoutwith a quantity called N[0,T ] (x). As it turns out, in many (deterministic) rough path estimates,as obtained in [FV10b] for instance, one may replace ωx (0, T ) by N[0,T ] (x). Doing so does notrequire to revisit the (technical) proofs of these rough path estimates, but rather to apply theexisting estimates repeatedly on the intervals of a carefully chosen partition of [0, T ]. The pointis that N[0,T ] (X) enjoys much better integrability than ωX (0, T ). Of course, this does not ruleout that for some rough paths x, N[0,T ] (x) ≈ ωx (0, T ), in agreement with the essentially optimalnature of exisiting rough path estimates in terms of ωx (0, T ). For instance, both quantities willscale like λ when p = 2 and x is the pure-area rough path, dilated by λ >> 1. Differently put,the point is that N[0,T ] (X (ω)) will be smaller than ωX(ω) (0, T ) for most realizations of X (ω).

The consequent focus on N[0,T ] rather than ”localized p-variation” aside, let us briefly enlistour contributions relative to [CLL].

(i) A technical condition “p > q [p]” is removed; this shows that the Cass, Litterer, Lyonsresults are valid assuming only “complementary Young regularity of the Cameron-Martinspace”, i.e. H → Cq−var where 1/p + 1/q > 1 and sample paths have finite p-variation,a natural condition, in particular in the context of Malliavin calculus, whose importancewas confirmed in a number of papers, [FV10a], [FO10], [CFV09], [CF10], see also [FV10b].

(ii) Their technical main result, Weibull tails of N[0,T ] (X) with shape parameter 2/q, hereX is a Gaussian rough path, remains valid for general rough paths obtained as image ofX under locally linear maps on (rough) path space. (This random rough paths may befar from Gaussian: examples of locally linear maps are given by rough integration and(solving) rough differential equations.)

Integrability RDEs

(iii) The arguments are adapted to deal with (random) linear (and also linear growth) roughdifferential equations (the solution maps here are not locally linear!) driven by X (ω). Asabove, it suffices that X is the locally linear image of a Gaussian rough path.

We conclude with two applications. First, we show how to recover log-Weibull tails for |J |,the Jacobian of a Gaussian RDE flow. (Afore-mentioned extended validity and some minorsharpening of the norm of |J | aside, this was the main result of [CLL].) Our point here is that[CLL] use somewhat involved (known) explicit estimates for the J in terms of the Gaussiandriving signal. In contrast, our “user-friendly” formulation allows for a simple step-by-stepapproach: recall that J solves dJ = J dM where M is a non-Gaussian driving rough path,obtained by solving an RDE / performing a rough integration. Since M is the locally linearimage of a Gaussian rough path, we can immediately appeal to (iii). As was pointed out recentlyby [HP], such estimates are - in combination with a Norris lemma for rough paths - the key toa non-Markovian Hoermander and then ergodic theory.Secondly, as a novel application, we consider (random) rough integrals of the form

∫G (X ) dX,

with G ∈ Lipγ−1,γ > p and establish Weibull tails with shape parameter 2/q, uniform overclasses of Gaussian process whose covariance satisfies a uniform variational estimate. A specialcase arises when X = Xε is taken, independently in each component, as solution to the stochasticheat equation on the 1D torus, u = uxx + W with hyper-viscosity term εuxxxx - as functionof the space variable, for fixed time. Complementary Young regularity is seen to hold withp > 2 and and q = 1 and we so obtain (and in fact, improve from exponential to Gaussianintegrability) the uniform in ε integrability estimate [Hai11, Theorem 5.1], a somewhat centraltechnical result whose proof encompasses almost a third of that paper.

3.1 Basic definitions

Definition 3.1.1. Let ω be a control. For α > 0 and [s, t] ⊂ [0, T ] we set

τ0 (α) = s

τi+1 (α) = inf u : ω (τi, u) ≥ α, τi (α) < u ≤ t ∧ t

and defineNα,[s,t] (ω) = sup n ∈ N∪0 : τn (α) < t .

When ω arises from a (homogenous) p-variation norm of a (p-rough) path, such as ωx =‖x‖pp-var;[·,·] or ωx := |||x|||pp-var;[·,·], detailed definitions are given later in the text, we shall alsowrite

Nα,[s,t] (x) := Nα,[s,t] (ωx) and Nα,[s,t] (x) := Nα,[s,t] (ωx) .

In fact, we will be in a situation where C−1ωx ≤ ωx ≤ Cωx for some constant C whichentails (cf. Lemma 3.1.3 below)

NαC,[·,·] (x) ≤ Nα,[·,·] (x) ≤ Nα/C,[·,·] (x) .

Furthermore, the precise value of α > 0 will not matter (c.f. Lemma 3.1.4 below) so that afactor C or 1/C is indeed inconsequential; effectively, this means that one can switch betweenN and N as one pleases.

We now study the scaling of Nα. Note that Nα,[s,t] (ω) 0 for α∞.

Lemma 3.1.2. Let ω be a control and λ > 0. Then (s, t) 7→ λω (s, t) is again a control and forall s < t,

Nα,[s,t] (λω) = Nα/λ,[s,t] (ω) .

Proof. Follows directly from the definition.

74

Basic definitions

Lemma 3.1.3. Let ω1, ω2 be two controls, s < t and α > 0. Assume that ω1 (u, v) ≤ Cω2 (u, v)holds whenever ω2 (u, v) ≤ α for a constant C. Then NCα,[s,t] (ω1) ≤ Nα,[s,t] (ω2).

Proof. It suffices to consider the case C = 1, the general case follows by the scaling of N . Set

τ j0 (α) = s

τ ji+1 (α) = infu : ωj

(τ ji , u

)≥ α, τ ji (α) < u ≤ t

∧ t

for j = 1, 2. It suffices to show that τ2i ≤ τ1

i holds for every i ∈ N. By induction over i: Fori = 0 this is clear. If τ2

i ≤ τ1i for some fixed i,

ω1

(τ1i , u)≤ ω2

(τ1i , u)≤ ω2

(τ2i , u)

whenever ω2

(τ2i , u)≤ α. Hence

infu

ω2

(τ2i , u)≥ α

≤ inf

u

ω1

(τ2i , u)≥ α

and therefore τ2

i+1 ≤ τ1i+1.

Lemma 3.1.4. Let ω be a control and 0 < α ≤ β. Then

Nα,[s,t] (ω) ≤ β

α

(2Nβ,[s,t] (ω) + 1

).

Proof. Set

ωα (s, t) := sup(ti)=D⊂[s,t]ω(ti,ti+1)≤α

∑ti

ω(ti, ti+1).

We clearly have ωα (s, t) ≤ ωβ (s, t) and

αNα,[s,t] (ω) =

Nα,[s,t](ω)−1∑i=0

ω(τi (α) , τi+1 (α)) ≤ ωα(s, t).

Finally, Proposition 4.6 in [CLL] shows that ωβ (s, t) ≤(2Nβ,[s,t] (ω) + 1

)β. (Strictly speaking,

Proposition 4.6 is formulated for a particular control ω, namely the control induced by the p-variation of a rough path. However, the proof only uses general properties of control functionsand the conclusion remains valid.)

Let x : [0, T ] → GN(Rd)

be a path. In the whole section, ‖·‖p−var denotes the p-variationnorm for such paths induced by the Carnot-Caratheodory metric; [FV10b]. Set ωx(s, t) =‖x‖pp−var;[s,t] and Nα,[s,t] (x) = Nα,[s,t] (ωx) (the fact that ωx is indeed a control is well-known;

c.f. [FV10b]).

Lemma 3.1.5. For any α > 0,

‖x‖p−var;[s,t] ≤ α1/p(Nα,[s,t] (x) + 1

)Proof. Let u = u0 < u1 < . . . < um = v. Note that

‖xu,v‖p =∥∥xu,u1 ⊗ xu1,u2 ⊗ . . .⊗ xum−1,v

∥∥p ≤ mp−1m−1∑i=0

∥∥xui,ui+1

∥∥p .75

Integrability RDEs

Let D be a dissection of [s, t] and (τj)Nα,[s,t](x)

j=0 = (τj (α))Nα,[s,t](x)

j=0 . Set D = D ∪ (τj)Nα,[s,t](x)

j=0 .Then, ∑

ti∈D

∥∥xti,ti+1

∥∥p ≤ (Nα,[s,t] (x) + 1

)p−1∑ti∈D

∥∥xti,ti+1

∥∥p≤

(Nα,[s,t] (x) + 1

)p−1Nα,[s,t](x)∑

j=0

‖x‖pp−var;[τj ,τj+1]

≤(Nα,[s,t] (x) + 1

)pα.

Taking the supremum over all partitions shows the claim.

3.2 Cass, Litterer and Lyons revisited

The basic object is a continuous d-dimensional Gaussian process, say X, realized as coor-dinate process on the (not-too abstract) Wiener space (E,H, µ) where E = C

([0, T ] ,Rd

)equipped with µ is a Gaussian measure s.t. X has zero-mean, independent components and

that Vρ-var

(R, [0, T ]2

), the ρ-variation in 2D sense of the covariance R of X, is finite for some

ρ ∈ [1, 2). From [FV10b, Theorem 15.33] it follows that we can lift the sample paths of X top-rough paths for any p > 2ρ and we denote this process by X, called the enhanced Gaussianprocess. We also assume that the Cameron-Martin space H has complementary Young regularityin the sense that H embeds continuously in Cq-var

([0, T ] ,Rd

)with 1

p + 1q > 1. Note q ≤ p for

µ is supported on the paths of finite p-variation. There are many examples of such a situation[FV10b], let us just note that fractional Brownian motion (fBM) with Hurst parameter H > 1/4falls in this class of Gaussian rough paths.

In this section, we present, in a self-contained fashion, the results [CLL]. In fact, we presenta slightly modified argument which avoids the technical condition ”p > q [p]” made in [CLL,Theorem 6.2, condition (3)] (this still applies to fBM with H > 1/4 but causes some disconti-nuities in the resulting estimates when H crosses the barrier 1/3). Our argument also gives aunified treatment for all p thereby clarifying the structure of the proof (in [CLL, Theorem 6.2]the cases [p] = 2, 3 are treated separately ”by hand”). That said, we clearly follow [CLL] intheir ingenious use of Borell’s inequality.

In the whole section, if not stated otherwise, for a p-rough path x, set

|||x|||p−var;[s,t] :=

[p]∑k=1

∥∥∥x(k)∥∥∥p/kp/k−var;[s,t]

1/p

.

Then |||·|||p−var is a homogeneous rough path norm. Recall that, as a consequence of Theorem7.44 in [FV10b], the norms |||·|||p−var and ||·||p−var are equivalent, hence there is a constant Csuch that

1

C|||·|||p−var ≤ ||·||p−var ≤ C |||·|||p−var . (3.1)

The map (s, t) 7→ ωx (s, t) = |||x|||pp−var;[s,t] is a control and we set Nα,[s,t] (x) = Nα,[s,t] (ωx).

Lemma 3.2.1. Assume that H has complementary Young regularity to X. Then for any a > 0,the set

Aa =|||X|||p−var;[0,T ] < a

has positive µ-measure. Moreover, if M ≥ Vρ−var

(R; [0, T ]2

), we have the lower bound

µ|||X|||p−var;[0,T ] < a

≥ 1− C

exp (a)

76

Cass, Litterer and Lyons revisited

where C is a constant only depending on ρ, p and M .

Proof. The support theorem for Gaussian rough paths ([FV10b, Theorem 15.60]) shows that

supp [X∗µ] = S[p] (H)

holds for p ∈ (2ρ, 4). Hence every neighbourhood of the zero-path has positive measure whichis the first statement. The general case follows from the a.s. estimate

|||S[p′] (X) |||p′−var ≤ |||S[p′] (X) |||p−var ≤ Cp,p′ |||X|||p−var (3.2)

which holds for every p ≤ p′, c.f. [FV10b, Theorem 9.5]. For the lower bound, recall that from[FV10b, Theorem 15.33] one can deduce that

E(

exp |||X|||p−var;[0,T ]

)≤ C (3.3)

for p ∈ (2ρ, 4) where C only depends on ρ, p and M . Using (3.2) shows that this actually holdsfor every p > 2ρ. Finally, by Chebychev’s inequality,

µ|||X|||p−var;[0,T ] < a

≥ 1− C

exp (a).

In the next theorem we cite the famous isoperimetric inequality due to C. Borell (for a proofc.f. [Led96, Theorem 4.3]).

Theorem 3.2.2 (Borell). Let (E,H, µ) be an abstract Wiener space and K denote the unit ballin H. If A ⊂ E is a Borell set with positive measure, then for every r ≥ 0

µ (A+ rK) ≥ Φ(Φ−1 (µ (A)) + r

)where Φ is the cumulative distribution function of a standard normal random variable, i.e.Φ = (2π)−1/2 ∫ ·

−∞ exp(−x2/2

)dx.

Corollary 3.2.3. Let f, g : E → [0,∞] be measurable maps and a, σ > 0 such that

Aa := x : f (x) ≤ a

has positive measure and let a ≤ Φ−1µ (Aa). Assume furthermore that there exists a null-set Nsuch that for all x ∈ N c and h ∈ H :

f (x− h) ≤ a⇒ σ ‖h‖H ≥ g (x) .

Then g has a Gaussian tail; more precisely, for all r > 0,

µ (x : g (x) > r) ≤ exp

(−(a+ r

σ

)22

).

Proof. W.l.o.g. σ = 1. Then

x : g (x) ≤ r =⋃h∈rK

x : ‖h‖H ≥ g (x)

⊃⋃h∈rK

x : f (x− h) ≤ a

=⋃h∈rK

x+ h : f (x) ≤ a

= Aa + rK.

By Theorem 3.2.2,µ (x : g (x) > r) ≤ µ (Aa + rKc) ≤ Φ (a+ r)

where Φ = 1− Φ. The claim follows from the standard estimate Φ (r) ≤ exp(−r2/2

).

77

Integrability RDEs

Proposition 3.2.4. Let X be a continuous d-dimensional Gaussian process, realized as coordi-nate process on (E,H, µ) where E = C

([0, T ] ,Rd

)equipped with µ is a Gaussian measure s.t.

X has zero-mean, independent components and that the covariance R of X has finite ρ-variationfor some ρ ∈ [1, 2). Let X be its enhanced Gaussian process with sample paths in a p-rough pathsspace, p > 2ρ. Assume H has complementary Young regularity, so that Cameron–Martin pathsenjoy finite q-variation regularity, q ≤ p and 1

p + 1q > 1. Then there exists a set E ⊂ E of full

measure with the following property: If

|||X (ω − h)|||p−var;[0,T ] ≤ α1/p (3.4)

for all ω ∈ E, h ∈ H and some α > 0 then

C |h|q−var;[0,T ] ≥ α1/p(Nβ,[0,T ] (X (ω))

)1/qwhere β = 2p [p]α and C depends only on p and q.

Proof. SetE = ω : Th (X (ω)) = X (ω + h) for all h ∈ H .

From [FV10b, Lemma 15.58] we know that E has full measure. Define the random partition(τi)∞i=0 = (τi (β))∞i=0 for the control ωX. Let h ∈ H and assume that (3.4) holds. We claim that

there is a constant Cp,q such that

Cp,q |h|q−var;[τi,τi+1] ≥ α1/p for all i = 0, . . . , Nβ,[0,T ] (X)− 1. (3.5)

The statement then follows from

Cqp,q |h|qq−var;[0,T ] ≥ C

qp,q

Nβ,[0,T ](X)−1∑i=0

|h|qq−var;[τi,τi+1] ≥ αq/pNβ,[0,T ] (X) .

To show (3.5), we first notice that for every i = 0, . . . , Nβ,[0,T ] (X)− 1,

β = |||X (ω) |||pp−var;[τi,τi+1] =

[p]∑k=1

∥∥∥X(k) (ω)∥∥∥p/kp/k−var;[τi,τi+1]

.

Fix i. Then there is a k ∈ 1, . . . , [p] such that∥∥X(k) (ω)

∥∥p/kp/k−var;[τi,τi+1]

≥ β[p] . Let D = (tj)

Mj=0

be any dissection of [τi, τi+1]. We define the vector

X(k) (ω) :=(X(k) (ω)t0,t1 , . . . ,X

(k) (ω)tM−1,tM

)and do the same for X(k) (ω − h) and for the mixed iterated integrals∫

∆k

dZi1 ⊗ . . .⊗ dZik where Zi =

X if i = 0h if i = 1

.

We have then

X(k) (ω − h) =∑

(i1,...,ik)∈0,1k(−1)i1+...+ik

∫∆k

dZi1 ⊗ . . .⊗ dZik

and by the triangle inequality,∣∣∣∣∫∆k

dh⊗ . . .⊗ dh∣∣∣∣lp/k

≥∣∣∣X(k) (ω)

∣∣∣lp/k−

∣∣∣X(k) (ω − h)∣∣∣lp/k

+∑

(i1,...,ik)∈0,1k0<i1+...+ik<k

∣∣∣∣∫∆k

dZi1 ⊗ . . .⊗ dZik∣∣∣∣lp/k

.(3.6)

78

Cass, Litterer and Lyons revisited

Since q < 2 and p ≥ q, we can use Young and super-additivity of |h|pq−var to see that∣∣∣∣∫∆k

dh⊗ . . .⊗ dh∣∣∣∣p/klp/k

≤ cp/kq,k

∑j

|h|pq−var;[tj ,tj+1]

≤ cp/kq,k |h|

pq−var;[τi,τi+1] .

For the mixed integrals one has for any u < v∣∣∣∣∣∫

∆ku,v

dZi1 ⊗ . . .⊗ dZik∣∣∣∣∣ ≤ ck,l,p,q |h|lq−var;[u,v] |||X (ω) |||k−lp−var;[u,v]

where l = i1 + . . .+ik (this follows from [FV10b, Theorem 9.26]). Hence we have, using Holder’sinequality and super-additivity∣∣∣∣∫

∆k

dZi1 ⊗ . . .⊗ dZik∣∣∣∣p/klp/k

≤ cp/kk,l,p,q

∑j

|h|lpk

q−var;[tj ,tj+1] |||X (ω) |||(k−l)pk

p−var;[tj ,tj+1]

≤ cp/kk,l,p,q

∑j

|h|pq−var;[tj ,tj+1]

l/k∑j

|||X (ω) |||pp−var;[tj ,tj+1]

k−lk

≤ cp/kk,l,p,q |h|

plk

q−var;[τi,τi+1] |||X (ω) |||p(k−l)k

p−var;[τi,τi+1]

and hence ∣∣∣∣∫∆k

dZi1 ⊗ . . .⊗ dZik∣∣∣∣lp/k

≤ ck,p,q |h|lq−var;[τi,τi+1] |||X (ω) |||k−lp−var;[τi,τi+1]

= ck,l,p,q |h|lq−var;[τi,τi+1] βk−lp .

By assumption, ∣∣∣X(k) (ω − h)∣∣∣lp/k≤ |||X (ω − h) |||kp−var;[0,T ] ≤ α

k/p.

Plugging this into (3.6) yields

cq,k |h|kq−var;[τi,τi+1] ≥∣∣∣X(k) (ω)

∣∣∣lp/k−

(αk/p +

k−1∑l=1

ck,l,p,q |h|lq−var;[τi,τi+1] βk−lp

).

Now we can take the supremum over all dissectionsD and obtain, using∥∥X(k) (ω)

∥∥p/k−var;[τi,τi+1]

≥(β[p]

)k/p,

cq,k |h|kq−var;[τi,τi+1] ≥∥∥∥X(k) (ω)

∥∥∥p/k−var;[τi,τi+1]

−

(αk/p +

k−1∑l=1


)

≥(β

[p]

)k/p−

(αk/p +

k−1∑l=1


)

=(

2k − 1)αk/p −

(k−1∑l=1

(2 [p]1/p

)k−lck,l,p,q |h|lq−var;[τi,τi+1] α

k−lp

).

By making constants larger if necessary, we may assume that there is a constant ck,p,q such that

|h|kq−var;[τi,τi+1] ≥(2k − 1

)ck,p,q

αk/p −

(k−1∑l=1

|h|lq−var;[τi,τi+1] αk−lp

).

79

Integrability RDEs

This implies that there is a constant Ck,p,q depending on ck,p,q such that

Ck |h|q−var;[τi,τi+1] ≥ α1/p.

Setting Cp,q = maxC1,p,q, . . . , C[p],p,q

finally shows (3.5).

Now we come to the main result.

Corollary 3.2.5. Let X be a centered Gaussian process in Rd with independent componentsand covariance RX of finite ρ-variation, ρ < 2. Consider the Gaussian p-rough paths X forp > 2ρ and assume that there is a continuous embedding

ι : H → Cq−var

where 1p + 1

q > 1 and let K ≥ ‖ι‖op. Then for every α > 0, Nα,[0,T ] (X) has a Weibull tail withshape 2/q. More precisely, there is a constant C = C (p, q) such that

µNα,[0,T ] (X) > r

≤ exp

−1

2

(a+

α1/pr1/q

CK

)2

for every r > 0 where a > −∞ is chosen such that

a ≤ Φ−1µ

|||X|||pp−var;[0,T ] ≤

α

2p [p]

.

Proof. Set

Aa =ω : |||X (ω) |||p−var;[0,T ] ≤ a1/p

.

Lemma 3.2.1 guarantees that Aa has positive measure for any a > 0. From Proposition 3.2.4we know that there is a set E of full measure such that whenever ‖X (ω − h)‖p−var;[0,T ] ≤ a1/p

for ω ∈ E, h ∈ H and a > 0 we have

a1/p(Nβ,[0,T ] (X)

)1/q ≤ cp,q |h|q−var;[0,T ] ≤ cp,q ‖ι‖op ‖h‖Hwhere β = 2p [p] a. Setting a = α/ (2p [p]), Corollary 3.2.3 shows that

µ(ω : Nα,[0,T ] (X (ω)) > r

)≤ exp

−( a√2

+α1/pr1/q

2√

2cp,q [p]1/p ‖ι‖op

)2

where a ≤ Φ−1µ (Aa).

Remark 3.2.6. Corollary 3.2.5 remains valid if one replaces Nα,[0,T ] by Nα,[0,T ] in the state-ment. This follows directly from (3.1) and Lemma 3.1.3 by putting the constant C of (3.1) inthe constant Cp,q of the respective corollaries.

Remark 3.2.7. In [FV10b, Proposition 15.7] it is shown that

|h|ρ−var ≤√Vρ−var

(RX ; [0, T ]2

)‖h‖H

holds for all h ∈ H. Hence in the regime ρ ∈ [1, 3/2) we can always choose q = ρ and the condi-tions of Corollary 3.2.5 are fulfilled. For the fractional Brownian motion with Hurst parameterH one can show that ρ = 1

2H and q > 1H+1/2 are valid choices (cf. [FV10b, chapter 15]) and

the results of Corollary 3.2.5 remain valid provided H > 1/4.

Remark 3.2.8. If X is a Gaussian rough paths, we know that ‖X‖p−var has a Gaussian tail(or a Weibull tail with shape parameter 2), e.g. obtained by a non-linear Fernique Theorem,cf. [FO10], whereas Corollary 3.2.5 combined with Lemma 3.1.5 only gives that ‖X‖p−var hasa Weibull tail with shape 2/q and thus the estimate is not sharp for q > 1. On the otherhand, Lemma 3.1.5 is robust and also available in situations where Fernique- (or Borell-) typearguments are not directly available, e.g. in a non-Gaussian setting.

80

Transitivity of the tail estimates under locally linear maps

3.3 Transitivity of the tail estimates under locally linear maps

Existing maps for rough integrals and RDE solutions suggest that we consider maps Ψ such that‖Ψ (x)‖p−var;I ≤ const. ‖x‖p−var;I uniformly over all intervals I ⊂ [0, T ] where ‖x‖p−var;I ≤ R,R > 0. More formally,

Definition 3.3.1. We call Ψ: Cp−var([0, T ] ;GN

(Rd))→ Cp−var

([0, T ] ;GM (Re)

)a locally

linear map if there is a R ∈ (0,∞] such that

‖Ψ‖R := infC>0

‖Ψ (x)‖p−var;[u,v] ≤ C ‖x‖p−var;[u,v] for all (u, v) ∈ ∆,x s.t. ‖x‖p−var;[u,v] ≤ R

is finite.

Remark 3.3.2. (i) For λ ∈ R, we denote by δλ the dilation map. Set (δλΨ) : x 7→ δλΨ (x).Then ‖·‖R is homogeneous w.r.t. dilation, e.g. ‖δλΨ‖R = |λ| ‖Ψ‖R.

(ii) If Ψ commutes with the dilation map δ modulo p-variation, e.g. ‖Ψ (δλx)‖p-var;I =‖δλΨ (x)‖p-var;I for any x, λ ∈ R and intervall I ⊂ [0, T ], we have ‖Ψ‖R = ‖Ψ‖∞ forany R > 0. An example of such a map is the Lyons lift map

SN : Cp−var(

[0, T ] ;G[p](Rd))→ Cp−var

([0, T ] ;GN

(Rd))

for which we have ‖SN‖∞ ≤ C (N, p) <∞, c.f. [FV10b, Theorem 9.5].

(iii) If φ : [0, T ] → φ [0, T ] ⊂ [0, T ] is a bijective, continuous and increasing function and x a

rough path, we set xφt = xφ(t) and call xφ a reparametrization of x. If Ψ commutes with

reparametrization modulo p-variation, e.g.∥∥Ψ(xφ)∥∥p−var;I =

∥∥∥Ψ (x)φ∥∥∥p−var;I

for any x,

φ and intervall I ⊂ [0, T ], we have

‖Ψ‖R := infC>0

‖Ψ (x)‖p−var;[0,T ] ≤ C ‖x‖p−var;[0,T ] for all x s.t. ‖x‖p−var;[0,T ] ≤ R

.

This follows by a standard reparametrization argument. Examples of such maps are roughintegration over 1-forms, e.g. x 7→

∫ ·0 ϕ (x) dx, and the Ito-Lyons map, e.g. Ψ (x)s,t = ys,t

where y solves dy = V (x) dx with initial condition y0 ∈ G[p] (Re). In this case,

‖Ψ‖∞ = supx

‖Ψ (x)‖p−var;[0,T ]

‖x‖p−var;[0,T ]

(where 0/0 := 0) and we find the usual operator norm. (Note that, however, we can notspeak of linear maps in this context since rough paths spaces are typically non-linear.)

(iv) Clearly, if ‖Ψ‖∞ <∞, ‖Ψ (X)‖p−var;[s,t] inherits the integrability properties of ‖X‖p−var;[s,t].However, for the most interesting maps, e.g. the Ito-Lyons map, we will not have‖Ψ‖∞ < ∞, but ‖Ψ‖R < ∞ for any finite R > 0. In a way, the purpose of this sec-tion is to show that one still has transitivity of integrability if one considers Nα,[s,t] (X)instead of ‖X‖p−var;[s,t].

Lemma 3.3.3. Let Ψ and Φ be locally linear maps with ‖Ψ‖R <∞ and ‖Φ‖R‖Ψ‖R <∞. ThenΨ Φ is again locally linear and

‖Φ Ψ‖R ≤ ‖Φ‖R‖Ψ‖R ‖Ψ‖R .

81

Integrability RDEs

Proof. Let ‖x‖p−var;[u,v] ≤ R. Then ‖Ψ (x)‖p−var;[u,v] ≤ ‖Ψ‖R ‖x‖p−var;[u,v] ≤ R ‖Ψ‖R whichimplies

‖Φ Ψ (x)‖p−var;[u,v] ≤ ‖Φ‖R‖Ψ‖R ‖Ψ (x)‖p−var;[u,v] ≤ ‖Φ‖R‖Ψ‖R ‖Ψ‖R ‖x‖p−var;[u,v] .

The interesting property of locally linear maps is formulated in the next proposition.

Proposition 3.3.4. Let Ψ: Cp−var([0, T ] ;GN

(Rd))→ Cp−var

([0, T ] ;GM (Re)

)be locally lin-

ear and ‖Ψ‖R <∞ for some R ∈ (0,∞]. Then

Nα‖Ψ‖pR,[s,t](Ψ (x)) ≤ Nα,[s,t] (x)

for any s < t and α ∈ (0, Rp].

Proof. Follows directly from Lemma 3.1.3.

3.1 Full RDEs

Consider the full RDE

dy = V (y) dx; y0 ∈ G[p] (Re) (3.7)

where x is a weak geometric p-rough path with values in G[p](Rd), V = (Vi)i=1,....d is a collection

of Lipγ-vector fields in Re where γ > p and y0 is the initial value. Theorem 10.36 and 10.38 in[FV10b] state that (3.7) possesses a unique solution y which is a weak geometric p-rough pathwith values in G[p] (Re).

Corollary 3.3.5. The Ito-Lyons map Ψ: x 7→ y is locally linear with

‖Ψ‖R ≤ K(‖V ‖Lipγ−1 ∨ ‖V ‖p

Lipγ−1 Rp−1)

(3.8)

for any R ∈ (0,∞) where K only depends on p and γ. Moreover, if ‖V ‖Lipγ−1 ≤ ν, then forany α > 0 there is a constant C = C (p, γ, ν, α) such that

Nα,[s,t] (y) ≤ C(Nα,[s,t] (x) + 1

)for any s < t.

Proof. (3.8) follows from the estimate (10.26) of [FV10b, Theorem 10.36]. From Proposition3.3.4 we obtain

Nβ,[s,t] (Ψ (x)) ≤ Nα,[s,t] (x)

where β = α ‖Ψ‖pα1/p . This already shows the claim if ‖Ψ‖p

α1/p ≤ 1. In the case ‖Ψ‖pα1/p > 1,

we conclude with Lemma 3.1.4.

3.2 Rough integrals

If x is a p-rough path and ϕ = (ϕi)i=1,...,d a collection of Lipγ−1(Rd,Re

)-maps, one can

define the rough integral ∫ϕ (x) dx (3.9)

as an element in Cp−var([0, T ] ;G[p] (Re)

)(c.f. [FV10b, chapter 10.6]).

82

Linear RDEs

Corollary 3.3.6. The map Ψ: x 7→ z, z given by the rough integral (3.9), is locally linear with

‖Ψ‖R ≤ K ‖ϕ‖Lipγ−1

(1 ∨Rp−1

)(3.10)

for any R ∈ (0,∞) where K only depends on p and γ. Moreover, if ‖ϕ‖Lipγ−1 ≤ ν, then for anyα > 0 there is a constant C = C (p, γ, ν, α) such that

Nα,[s,t] (z) ≤ C(Nα,[s,t] (x) + 1

)for any s < t.

Proof. (3.10) follows from [FV10b, Theorem 10.47]. One proceeds as in the proof of Corollary3.3.5.

3.4 Linear RDEs

For a p-rough path x, consider the full linear RDE

dy = V (y) dx; y0 ∈ G[p] (Re) (3.11)

where V = (Vi)i=1,...,d is a collection of linear vector fields of the form Vi (z) = Aiz + bi, Ai aree× e matrices and bi ∈ Re. It is well-known (e.g. [FV10b, section 10.7]) that in this case (3.11)has a unique solution y. Unfortunately, the map Ψ: x 7→ y is not locally linear in the sense ofDefinition 3.3.1 and our tools of the former section do not apply. However, we can do a moredirect analysis and obtain a different transitivity of the tail estimates.

Let ν be a bound on maxi (|Ai|+ |bi|) and set y = π1 (y). In [FV10b, Theorem 10.53] onesees that there is a constant C depending only on p such that

‖ys,t‖ ≤ C (1 + |ys|) ν ‖x‖p−var;[s,t] exp(Cνp ‖x‖pp−var;[s,t]

)(3.12)

holds for all s < t ∈ [0, T ]. (Strictly speaking, we only find the estimate for (s, t) = (0, 1), thegeneral case follows by reparametrization.) We start with an estimate for the supremum normof y.

Lemma 3.4.1. For any α > 0 there is a constant C = C (p, ν, α) such that

|y|∞;[s,t] ≤ C (1 + |ys|) exp(CNα;[s,t] (x)

)holds for any s < t.

Proof. From (3.12) we have

|yu,v| ≤ C (1 + |yu|) ν ‖x‖p−var;[u,v] exp(Cνp ‖x‖pp−var;[u,v]

)(3.13)

for any u < v ∈ [s, t]. From |yu,v| = |ys,v − ys,u| ≥ |ys,v| − |ys,u| we obtain

|ys,v| ≤ C (1 + |yu|) ν ‖x‖p−var;[u,v] exp(Cνp ‖x‖pp−var;[u,v]

)+ |ys,u|

≤ C (1 + |ys|+ |ys,u|) expCνp ‖x‖pp−var;[u,v]

by making C larger. Now let s = τ0 < . . . < τN < τM+1 = u ≤ t with M ≥ 0. By induction,one sees that

|ys,u| ≤ CM+1 ((M + 1) (1 + |ys|)) exp

C

M∑i=0

νp ‖x‖pp−var;[τi,τi+1]

≤ CM+1 (1 + |ys|) exp

C

M∑i=0

νp ‖x‖pp−var;[τi,τi+1]

.

83

Integrability RDEs

This shows that for every u ∈ [s, t],

|ys,u| ≤ C(Nα;[s,t](x)+1) (1 + |ys|) exp(Cνpα

(Nα;[s,t] (x) + 1

))= (1 + |ys|) exp

(log (C) + Cνpα)

(Nα;[s,t] (x) + 1

)and hence

supu∈[s,t]

|ys,u| ≤ C (1 + |ys|) exp(CNα;[s,t] (x)

)for a constant C = C (p, ν, α) and therefore also

|y|∞;[s,t] ≤ C (1 + |ys|) exp(CNα;[s,t] (x)

).

Corollary 3.4.2. Let α > 0. Then there is a constant C = C (p, ν, α) such that

Nα,[s,t] (y) ≤ C (1 + |ys|)p exp(CNα;[s,t] (x)

)for any s < t.

Proof. Using (3.12) we can deduce that

‖yu,v‖ ≤ C(

1 + |y|∞;[s,t]

)ν ‖x‖p−var;[u,v] exp

(Cνp ‖x‖pp−var;[u,v]

)holds for any u < v ∈ [s, t] and hence also

‖y‖p−var;[u,v] ≤ C(

1 + |y|∞;[s,t]

)ν ‖x‖p−var;[u,v] exp

(Cνp ‖x‖pp−var;[u,v]

)for any u < v ∈ [s, t]. Now take u < v ∈ [s, t] such that ‖x‖pp−var;[u,v] ≤ α. We then have

‖y‖pp−var;[u,v] ≤ C ‖x‖pp−var;[u,v]

whereC = Cp

(1 + |y|∞;[s,t]

)pνp exp (pCνpα) .

From Lemma 3.1.3,NCα,[s,t] (y) ≤ Nα,[s,t] (x) .

If C ≤ 1, this already shows the claim. For C > 1, we use Lemma 3.1.4 and Lemma 3.4.1to see that

Nα,[s,t] (y) ≤(

2NCα,[s,t] (y) + 1)C

≤ C(Nα,[s,t] (x) + 1

) (1 + |y|∞;[s,t]

)p≤ C (1 + |ys|)p exp

(CNα,[s,t] (x)

).

Remark 3.4.3 (Unbounded vector fields). Let x be a p-rough path. Consider a collectionV = (Vi)1≤i≤d of locally Lipγ−1-vector fields on Re, γ ∈ (p, [p] + 1), such that Vi are Lips-

chitz continuous and the vector fields V [p] =(Vi1 , . . . , Vi[p]

)i1,...,i[p]∈1,...,d

are (γ − [p])-Holder

continuous. Then the RDE

dy = V (y) dx; y0 ∈ G[p] (Re)

84

Applications in stochastic analysis

has a unique solution (c.f. [FV10b, Exercise 10.56] and the solution thereafter and [Lej09]).Moreover, in [FV10b] it is shown that

‖y0,1‖ ≤ C (1 + |y0|) ν ‖x‖p−var;[0,1] exp(Cνp ‖x‖pp−var;[0,1]

)where C = C (p, γ) and ν is a bound on

∣∣V [p]∣∣1/[p](γ−[p])-Hol

∨ supy,z|V (y)−V (z)||y−z| . This shows that

Lemma 3.4.1 and Corollary 3.4.2 apply for y, hence for any α > 0 there is a constant C =C (p, γ, ν, α) such that

Nα,[s,t] (y) ≤ C (1 + |ys|)p exp(CNα;[s,t] (x)

)for all s < t in this case.

3.5 Applications in stochastic analysis

5.1 Tail estimates for stochastic integrals and solutions of SDEs driven by Gaussiansignals

We now apply our results to solutions of SDEs and stochastic integrals driven by Gaussiansignals, i.e. a Gaussian rough path X. Remark that all results here may be immediatelyformulated for SDEs and stochastic integrals driven by random rough paths as along as suitablequantitative Weibull-tail estimate for Nα,[0,T ] (X) are assumed.

We first consider the non-linear case:

Proposition 3.5.1. Let X be a centered Gaussian process in Rd with independent componentsand covariance RX of finite ρ-variation, ρ < 2. Consider the Gaussian p-rough paths X forp > 2ρ and assume that there is a continuous embedding

ι : H → Cq−var

where 1p + 1

q > 1. Let Y : [0, T ]→ Re be the pathwise solution of the stochastic RDE

dY = V (Y ) dX; Y0 ∈ Re

where V = (Vi)i=1,....d is a collection of Lipγ-vector fields in Re with γ > p. Moreover, letZ : [0, T ]→ Re be the stochastic integral given by

Zt = π1

(∫ t

0ϕ (X) dX

)where ϕ = (ϕi)i=1,...,d is a collection of Lipγ−1

(Rd,Re

)-maps, γ > p. Then both ‖Y ‖p−var;[0,T ]

and ‖Z‖p−var;[0,T ] have Weibull tails with shape parameter 2/q. More precisely, if K ≥ ‖ι‖op,

M ≥ Vρ−var

(R; [0, T ]2

)and ν ≥ ‖V ‖Lipγ−1 there is a constant η = η (p, q, ρ, γ, ν,K,M) > 0

such that

P(‖Y ‖p−var;[0,T ] > r

)≤ 1

ηexp

(−ηr2/q

)for all r ≥ 0

and the same holds for ‖Z‖p−var;[0,T ] if ν ≥ ‖ϕ‖Lipγ−1 instead. In particular, ‖Y ‖p−var;[0,T ] and‖Z‖p−var;[0,T ] have finite exponential moments as long as q < 2.

Proof. From 3.2.1 we know that there is a α = α (ρ, p,M) such that

P

|||X|||pp−var;[0,T ] ≤

α

2p [p]

≥ 1

2.

85

Integrability RDEs

Hence, by Corollary 3.2.5, applied with a = Φ−1(

12

)= 0, and the remark thereafter,

PNα,[0,T ] (X) > r

≤ exp

−1

2

(α1/pr1/q

c1

)2 for all r ≥ 0

with c1 = c1 (p, q,K,M). Corollary 3.3.5 shows that there is a constant c2 = c2 (p, q,K,M, γ, ν)such that also

PNα,[0,T ] (Y) > r

≤ c2 exp

−r

2/q

c2

for all r ≥ 0.

From Lemma 3.1.5 we see that

‖Y ‖p−var;[0,T ] ≤ ‖Y‖p−var;[0,T ] ≤ α1/p(Nα,[0,T ] (Y) + 1

)which shows the claim for ‖Y ‖p−var;[0,T ]. The same holds true for ‖Z‖p−var;[0,T ] by using

Corollary 3.3.6.

Remark 3.5.2. In the Brownian motion case (q = 1), we recover the well-known fact thatsolutions Y of the Stratonovich SDE

dY = V (Y ) dB; Y0 ∈ Re

have Gaussian tails at any fixed point Yt provided V is sufficiently smooth. We also recover thatthe Stratonovich integral ∫ t

0ϕ (B) dB

has finite Gaussian tails for every t ≥ 0, ϕ sufficiently smooth.

Proposition 3.5.3. Let X be as in Proposition 3.5.1. Let Y : [0, T ] → Re be the pathwisesolution of the stochastic linear RDE

dY = V (Y ) dX; Y0 ∈ Re

where V = (Vi)i=1,...,d is a collection of linear vector fields of the form Vi (z) = Aiz + bi, Ai are

e× e matrices and bi ∈ Re. Then log(‖Y ‖p−var;[0,T ]

)has a Weibull tail with shape 2/q. More

precisely, if K ≥ ‖ι‖op, M ≥ Vρ−var

(R; [0, T ]2

)and ν ≥ maxi (|Ai|+ |bi|) there is a constant

η = η (p, q, ρ, ν,K,M) > 0 such that

P(

log(‖Y ‖p−var;[0,T ]

)> r)≤ 1

ηexp

(−ηr2/q

)for all r ≥ 0

In particular, ‖Y ‖p−var;[0,T ] has finite Ls-moments for any s > 0 provided q < 2.

Proof. Same as for Proposition 3.5.1 using Corollary 3.4.2.

Remark 3.5.4. In the case q = 1, which covers Brownian driving signals, we have log-normaltails. This is in agreement with trivial examples such as the standard Black-Scholes model inwhich the stock price St is log-normally distributed.

Remark 3.5.5. The same conclusion holds for unbounded vector fields as seen in remark 3.4.3.

86


5.2 The Jacobian of the solution flow for SDEs driven by Gaussian signalsLet x : [0, T ] → Rd be smooth and let V =

(V 1, . . . , V d

): Re → Re be a collection of

vector fields. We can interpret V as a function V : Re → L(Rd,Re

)with derivative DV : Re →

L(Re, L

(Rd,Re

)) ∼= L(Rd,End (Re)

). It is well-known that for sufficiently smooth V , the ODE

dy = V (y) dx

has a solution for every starting point y0 and the solution flow y0 → Ut←0 (y0) = yt is (Frechet)differentiable. We denote its derivative by Jxt←0 (y0) = DUt←0 (·) |·=y0 . Moreover, for fixed y0,the Jacobian Jt = Jxt←0 (y0) is given as the solution of the linear ODE

dJt = dMt · Jt; J0 = Id

where Mt ∈End(Re) is given by the integral

Mt =

∫ t

0DV (ys) dxs. (3.14)

If x is a p-rough path, one proceeds in a similar fashion. First, in order to make sense of(3.14) if x and y are rough paths, one has to define the joint rough path (x,y) = z ∈Cp−var

([0, T ] , G[p]

(Rd ⊕ Re

))first. To do so, one defines z as the solution of the full RDE

dz = V (z) dx; z0 = exp (0, y0) .

where V = (Id, V ). Then, one defines M ∈ Cp−var([0, T ] , G[p] (Re×e)

)as the rough integral

Mt =

∫ t

0φ (z) dz

where φ : Rd ⊕ Re → L(Rd ⊕ Re,End (Re)

)is given by φ (x, y) (x′, y′) = DV (y) (x′) for all

x, x′ ∈ Rd and y, y′ ∈ Re. Finally, one obtains Jxt = Jx

t←0 (y0) as the solution of the linear RDE

dJxt = dMt · Jx

t ; J0 = Id.

All this can be made rigorous; for instance, see [FV10b, Theorem 11.3]. Next, we give analternative proof of the main result of [CLL], slightly sharpened in the sense that we considerthe p-variation norm instead of the supremum norm.

Proposition 3.5.6. Let X be a centered Gaussian process in Rd with independent componentsand covariance RX of finite ρ-variation, ρ < 2. Consider the Gaussian p-rough paths X forp > 2ρ and assume that there is a continuous embedding

ι : H → Cq−var

where 1p + 1

q > 1. Then log(∥∥JX

·←0 (y0)∥∥p−var;[0,T ]

)has a Weibull tail with shape 2/q. In

particular, if q < 2, this implies that∥∥JX·←0 (y0)

∥∥p−var;[0,T ]

has finite Lr-moments for any r > 0.

Proof. From Corollary 3.2.5 we know that N1,[0,T ] (X) has a Weibull tail with shape 2/q. Com-bining the Corollaries 3.3.5, 3.3.6 and 3.4.2 shows that there is a constant C such that

log(N1,[0,T ]

(JX·←0 (y0)

)+ 1)≤ C

(N1;[0,T ] (X) + 1

).

From Lemma 3.1.5 we know that∥∥JX·←0 (y0)

∥∥p−var;[0,T ]

≤∥∥JX·←0 (y0)

∥∥p−var;[0,T ]

≤ N1,[0,T ]

(JX·←0 (y0)

)+ 1.

87

Integrability RDEs

5.3 An example from rough SPDE theory

In situations where one performs a change of measure to an equivalent measure on a pathspace, one often has to make sense of the exponential moments of a stochastic integral, i.e. toshow that

E

(exp

∫G (X) dX +

∫F (X) dt

)(3.15)

is finite for a given process X and some suitable maps G and F . The second integral is oftentrivially handled (say, when F is bounded) and thus take F = 0 in what follows. Varioussituations in the literature (e.g. [Hai11], [CDFO]) require to bound (3.15) uniformly over afamily of processes, say (Xε : ε > 0). We will see in this section that our results are perfectlysuited for doing this.

In the following, we study the situation of [Hai11, section 4]. Here ψε = ψε (t, x;ω) is thestationary (in time) solution to the damped stochastic heat equation with hyper-viscosity ofparameter ε > 0,

dψε = −ε2∂xxxxψε dt+ (∂xx − 1)ψε dt+√

2 dWt

where W is space-time white noise, a cylindrical Wiener process over L2 (T) where T denotesthe torus, say [−π, π] with periodic boundary conditions. Following [Hai11] we fix t, so that the”spatial” interval [−π, π] plays the role of our previous ”time-horizon” [0, T ]. Note thatx 7→ψε (x, t) is a centered Gaussian process on T, with independent components and covariancegiven by

E (ψε (x, t)⊗ ψε (y, t)) = Rε (x, y) I = Kε (x− y) I

where Kε (x) is proportional to ∑k∈Z

cos (kx)

1 + k2 + ε2k4.

As was pointed out by Hairer, it can be very fruitful in a non-linear SPDE context to considerlimε→0 ψ

ε (t, ·, ω) as random spatial rough path. To this end, it is stated (without proof) in[Hai11] that the covariance of ψε has fnite ρ-variation in 2D sense, ρ > 1, uniformly in ε. Infact, we can show something slightly stronger. Following [Hai11], ψε is C1 in x for every ε > 0,and can be seen as p-rough path, any p > 2, when ε = 0.

Lemma 3.5.7. The map T2 3 (x, y) 7→ Rε (x, y) has finite 1-variation in 2D sense, uniformlyin ε. That is,

M := supε≥0

V1−var(Rε;T2

)<∞.

Proof. By lower semi-continuity of variation norms under pointwise convergence, it suffices toconsider ε > 0. (Alternatively, the case ε = 0 is treated explicitly in [Hai11]). We then notethat ∑

k∈Z

cos (kx)

1 + ε2k2=π cosh

(1ε (|x| − π)

)ε sinh

(πε

)in L2 (T) as may be seen by Fourier expansion on [−π, π] of the function x 7→ cosh

(1ε (|x| − π)

).

Since |∂x,yRε (x, y)| = |K ′′ε (x− y)| we have

V1−var(Rε;T2

)=

∫T2

|∂x,yRε (x, y)| dx dy =

∫T2

∣∣K ′′ε (x− y)∣∣ dx dy

88


On the other hand,

∣∣K ′′ε (x)∣∣ ≤ ∣∣∣∣∣∑

k∈Z

(1

1 + ε2k2− k2

1 + k2 + ε2k4

)cos (kx)

∣∣∣∣∣+π cosh

(1ε (|x| − π)

)ε sinh

(πε

)=

∣∣∣∣∣∣1 +∑k 6=0

1

k2

cos (kx)

(1 + ε2k2) (1/k2 + 1 + ε2k2)

∣∣∣∣∣∣+π cosh

(1ε (|x| − π)

)ε sinh

(πε

)≤ 1 +

∑k 6=0

1

k2+π cosh

(1ε (|x| − π)

)ε sinh

(πε

)≤ 1 +

π2

3+π cosh

(1ε (|x| − π)

)ε sinh

(πε

) .

Hence∫T2

∣∣K ′′ε (x− y)∣∣ dx dy ≤ (2π)2

(1 +

π2

3

)+

π

ε sinh(πε

) ∫T2

cosh

(1

ε(|x− y| − π)

)dx dy.

We leave to the reader to see that the final integral is bounded, independent of ε. For instance,introduce z = x− y as new variable so that only

π

ε sinh(πε

) ∫ 2π

−2πcosh

(1

ε(|z| − π)

)dz

= 4π

ε sinh πε

∫ π

0cosh

(zε

)dz

needs to be controlled. Using cosh ≈ sinh ≈ exp for large arguments (or integrating explicitly...) we get

1

ε sinh(πε

) ∫ π

0cosh

(zε

)dz ≈

∫ π

0

exp(zε

)ε exp

(πε

) dz=

exp(πε

)− 1

exp(πε

)≤ 1

We then have the following sharpening of [Hai11, Theorem 5.1].

Theorem 3.5.8. Fix γ > 2 and p ∈ (2, γ). Assume G = (Gi)i=1,...,d is a collection of

Lipγ−1(Rd,Re

)maps. Then for some constant η = η

(γ, p, ‖G‖Lipγ−1

)> 0 we have the uniform

estimate

supt∈[0,∞)

supε≥0

E

exp

(η

∣∣∣∣∫TG (ψε (x, t)) dxψ

ε (x, t)

∣∣∣∣2)

<∞.

(When ε > 0, ψε is known to be C1 in x so that we deal with Riemann–Stieltjes integrals, whenε = 0, the integral is understood in rough path sense.)

Proof. By stationarity in time of ψε (·, t), uniformity in t is trivial. Note that the Riemann–Stieltjes integral ∫

TG (ψε (x, t)) dxψ

ε (x, t)

89

Integrability RDEs

can also be seen as rough integral where the integrator is given by the ”smooth” rough path(ψε,∫ψε ⊗ dxψε

)when ε > 0. For ε = 0, the above integral is a genuine rough integral, the

existence of a canonical lift of ψ0 (·, t) to a geometric rough path is a standard consequence(cf. [FV10a, FV10b]) of finite 1-variation of R0, the covariance function of ψ0 (·, t). After theseremarks,

supε≥0

E

exp

(C

∣∣∣∣∫TG (ψε (x, t)) dxψ

ε (x, t)

∣∣∣∣2)

<∞

is an immediate application of Lemma 3.5.7 and Proposition 3.5.1.

90

4

A simple proof of distance boundsfor Gaussian rough paths

The intersection between rough path theory and Gaussian processes has been an active researcharea in recent years ([FV10a], [FV10b], [Hai11]). The central idea of rough paths, as realized byLyons ([Lyo98]), is that the key properties needed for defining integration against an irregularpath do not only come from the path itself, but from the path together with a sequence ofiterated integrals along the path, namely

Xns,t =

∫s<u1<···<un<t

dXu1 ⊗ . . .⊗ dXun . (4.1)

In particular, Lyons extension theorem shows that for paths of finite p-variation, the first bpclevels iterated integrals determine all higher levels. For instance, if p = 1, the path has boundedvariation and the higher iterated integrals coincide with the usual Riemann-Stieltjes integrals.However, for p ≥ 2, this is not true anymore and one has to say what the second (and possiblyhigher) order iterated integrals should be before they determine the whole rough path.

Lyons and Zeitouni ([LZ99]) were the first to study iterated Wiener integrals in the sense ofrough paths. They provide sharp exponential bounds on the iterated integrals of all levels bycontrolling the variation norm of the Levy area. The case of more general Gaussian processeswere studied by Friz and Victoir in [FV10a] and [FV10b]. They showed that if X is a Gaussianprocess with covariance of finite ρ-variation for some ρ ∈ [1, 2), then its iterated integrals in thesense of (4.1) can be defined in a natural way and we can lift X to a Gaussian rough path X.

In the recent work [FR], Friz and the first author compared the two lift maps X and Y forthe joint process (X,Y ). It was shown that their average distance in rough paths topology can

be controlled by the value supt |Xt−Yt|ζL2 for some ζ > 0, and a sharp quantitative estimate forζ was given. In particular, it was shown that considering both rough paths in a larger roughpaths space (and therefore in a different topology) allows for larger choices of ζ. Using this,the authors derived essentially optimal convergence rates for Xε → X in rough paths topologywhen ε→ 0 where Xε is a suitable approximation of X.

In order to prove this result, sharp estimates of |Xns,t −Yn

s,t| need to be calculated on every

level n. Under the assumption ρ ∈ [1, 32), the sample paths of X and Y are p-rough paths for

any p > 2ρ, hence we can always choose p < 3 and therefore the first two levels determine theentire rough path. Lyons’ continuity theorem then suggests that one only needs to give sharpestimates on level 1 and 2; the estimates on the higher levels can be obtained from the lowerlevels through induction. On the other hand, interestingly, one additional level was estimated”by hand” in [FR] before performing the induction. To understand the necessity of computingthis additional term, let us note from [Lyo98] that the standard distance for two deterministic

Simple proof distance bounds

p-rough paths takes the form of the smallest constant Cn such that

|Xns,t −Yn

s,t| ≤ Cnεω(s, t)np , n = 1, · · · , bpc

holds for all s < t where ω is a control function to be defined later. The exponent on the controlfor the next level is expected to be

n+ 1

p=bpc+ 1

p> 1, (4.2)

so when one repeats Young’s trick of dropping points in the induction argument (the key idea ofthe extension theorem), condition (4.2) will ensure that one can establish a maximal inequalityfor the next level. However, in the current problem where Gaussian randomness is involved, theL2 distance for the first b2ρc iterated integrals takes the form

|Xns,t −Yn

s,t|L2 < Cnεω(s, t)12γ

+n−12ρ , n = 1, 2, ρ ∈ [1,

3

2),

where γ might be much larger than ρ. Thus, the ’n− 1’ in the exponent leaves condition (4.2)unsatisfied, and one needs to compute the third level by hand before starting induction on n.

In this article, we resolve the difficulty by moving part of ε to fill in the gap in the controlso that the exponent for the third level control reaches 1. In this way, we obtain the third levelestimate merely based on the first two levels, and it takes the form

|X3s,t −Y3

s,t|L2 < C3εηω(s, t),

where η ∈ (0, 1], and its exact value depends on γ and ρ. We see that there is a 1− η reductionin the exponent of ε, which is due to the fact that it is used to compensate the control exponent.This interplay between the ’rate’ exponent and the control exponent can be viewed as an analogyto the relationship between time and space regularities for solutions to SPDEs. We will makethe above heuristic argument rigorous in section 4. We also refer to the recent work [LX12] forthe situation of deterministic rough paths.

Our main theorem is the following.

Theorem 4.0.1. Let (X,Y ) = (X1, Y 1, · · · , Xd, Y d) : [0, T ] → Rd+d be a centered Gaussianprocess on a probability space (Ω,F , P ) where (Xi, Y i) and (Xj , Y j) are independent for i 6= j,with continuous sample paths and covariance function R(X,Y ) : [0, T ]2 7→ R2d×2d. Assume further

that there is a ρ ∈ [1, 32) such that the ρ-variation of R(X,Y ) is bounded by a finite constant K.

Let γ ≥ ρ such that 1γ + 1

ρ > 1. Then, for every σ > 2γ, N ≥ bσc, q ≥ 1 and every δ > 0 smallenough, there exists a constant C = C(ρ, γ, σ,K, δ, q) such that

(i) If 12γ + 1

ρ > 1, then

|ρNσ−var(X,Y)|Lq ≤ C supt∈[0,T ]

|Xt − Yt|1− ρ

γ

L2 .

(ii) If 12γ + 1

ρ ≤ 1, then

|ρNσ−var(X,Y)|Lq ≤ C supt∈[0,T ]

|Xt − Yt|3−2ρ−δL2 .

The proof of this theorem will be postponed to subsection 4.2.2 after we have establishedall the estimates needed.

92

2D variation and Gaussian rough paths

Remark 4.0.2. We emphasize that the constant C in the above theorem depends on the process(X,Y ) only through the parameters ρ and K.

This chapter is structured as follows. In section 4.1, we introduce the class of Gaussian pro-cesses which possess a lift to Gaussian rough paths and estimate the difference of two Gaussianrough paths on level one and two. Section 4.2 is devoted to the proof of the main theorem. Wefirst obtain the third level estimate directly from the first two levels, which requires a technicalextension of Lyons’ continuity theorem, and justify the heuristic argument above rigorously. Allhigher level estimates are then obtained with the induction procedure in [Lyo98], and the claimof the main theorem follows. In section 4.3, we give two applications of our main theorem. Thefirst one deals with convergence rates for Wong-Zakai approximations in the context of roughdifferential equations. The second example shows how to derive optimal time regularity for thesolution of a modified stochastic heat equation seen as an evolution in rough paths space.

Notations. Throughout the chapter, C,Cn, Cn(ρ, γ) will denote constants depending oncertain parameters only, and their actual values may change from line to line.

4.1 2D variation and Gaussian rough paths

If I = [a, b] is an interval, a dissection of I is a finite subset of points of the form a = t0 <. . . < tm = b. The family of all dissections of I is denoted by D(I).

Let I ⊂ R be an interval and A = [a, b]× [c, d] ⊂ I × I be a rectangle. Recall the followingdefinitions: If f : I × I → V is a function, mapping into a normed vector space V , we define therectangular increment f(A) by setting

f(A) := f

(a, bc, d

):= f

(bd

)− f

(ad

)− f

(bc

)+ f

(ac

).

Definition 4.1.1. Let p ≥ 1 and f : I × I → V . For [s, t]× [u, v] ⊂ I × I, set

Vp(f ; [s, t]× [u, v]) :=

sup(ti)∈D([s,t])(t′j)∈D([u,v])

∑ti,t′j

∣∣∣∣f ( ti, ti+1

t′j , t′j+1

)∣∣∣∣p

1p

.

If Vp(f, I × I) <∞, we say that f has finite (2D) p-variation. We also define

V∞(f ; [s, t]× [u, v]) := supσ,τ∈[s,t]µ,ν∈[u,v]

∣∣∣∣f ( σ, τµ, ν

)∣∣∣∣

Lemma 4.1.2. Let f : I × I → V be a continuous map and 1 ≤ p ≤ p′ < ∞. Assume that fhas finite p-variation. Then for every [s, t]× [u, v] ⊂ I × I we have

Vp′(f ; [s, t]× [u, v]) ≤ V∞(f ; [s, t]× [u, v])1− p

p′ Vp(f ; [s, t]× [u, v])pp′

93


Proof. Let (ti) ∈ D([s, t]) and (t′j) ∈ D([u, v]). Then,

∑ti,t′j

∣∣∣∣f ( ti, ti+1

t′j , t′j+1

)∣∣∣∣p′ ≤ V∞(f ; [s, t]× [u, v])p′−p∑ti,t′j

∣∣∣∣f ( ti, ti+1

t′j , t′j+1

)∣∣∣∣p .Taking the supremum over all partitions gives the claim.

Lemma 4.1.3. Let f : I × I → R be continuous with finite p-variation. Choose p′ such thatp′ ≥ p if p = 1 and p′ > p if p > 1. Then there is a control ω and a constant C = C(p, p′) suchthat

Vp′(f ; J × J) ≤ ω(J)1p′ ≤ CVp(f ; J × J)

holds for every interval J ⊂ I.

Proof. Follows from [FV11, Theorem 1].

Let X = (X1, . . . , Xd) : I → Rd be a centered, stochastic process. Then the covariancefunction RX(s, t) := CovX(s, t) = E(Xs ⊗Xt) is a map RX : I × I → Rd×d and we can ask forits ρ-variation (we will use the letter ρ instead of p in this context). Clearly, RX has finite ρ-variation if and only if for every i, j ∈ 1, . . . , d the map s, t 7→ E(Xi

sXjt ) has finite ρ-variation.

In particular, if Xi and Xj are independent for i 6= j, RX has finite ρ-variation if and only if RXi

has finite ρ-variation for every i = 1, . . . , d. In the next example, we calculate the ρ-variationfor the covariances of some well-known real valued Gaussian processes. In particular, we willsee that many interesting Gaussian processes have a covariance of finite 1-variation.

Example 4.1.4

(i) Let X = B be a Brownian motion. Then RB(s, t) = mins, t and thus, forA = [s, t]× [u, v],

|R(A)| = |(s, t) ∩ (u, v)| =∫

[s,t]×[u,v]δx=y dx dy.

This shows that RB has finite 1-variation on any interval I.

(ii) More generally, let f : [0, T ]→ R be a left-continuous, locally bounded function. Set

Xt =

∫ t

0f(r) dBr.

Then, for A = [s, t] ∩ [u, v] we have by the Ito isometry,

RX(A) = E

[∫[s,t]

f dB

∫[u,v]

f dB

]=

∫[s,t]×[u,v]

δx=yf(x)f(y) dx dy

which shows that RX has finite 1-variation.

(iii) Let X be an Ornstein-Uhlenbeck process, i.e. X is the solution of the SDE

dXt = −θXt dt+ σ dBt (4.3)

for some θ, σ > 0. If we claim that X0 = 0, one can show that X is centered, Gaussianand a direct calculation shows that the covariance of X has finite 1-variation on anyinterval [0, T ]. The same is true considering the stationary solution of (4.3) instead.

94


(iv) If X is a continuous Gaussian martingale, it can be written as a time-changed Brownianmotion. Since the ρ-variation of its covariance is invariant under time-change, X hasagain a covariance of finite 1-variation.

(v) If X : [0, T ]→ R is centered Gaussian with X0 = 0, we can define a Gaussian bridge by

XBridge(t) = Xt − tXT

T.

One can easily show that if the covariance of X has finite ρ-variation, the same is truefor XBridge. In particular, Brownian bridges have finite 1-variation.

Next, we cite the fundamental existence result about Gaussian rough paths. For a proof, cf.[FV10a] or [FV10b, Chapter 15].

Theorem 4.1.5 (Friz, Victoir). Let X : [0, T ] → Rd be a centered Gaussian process with con-tinuous sample paths and independent components. Assume that there is a ρ ∈ [1, 2) suchthat Vρ(RX ; [0, T ]2) < ∞. Then X admits a lift X to a process whose sample paths are geo-metric p-rough paths for any p > 2ρ, i.e. with sample paths in C0,p−var([0, T ], Gbpc(Rd)) andπ1(Xs,t) = Xt −Xs for any s < t.

In the next proposition, we give an upper L2-estimate for the difference of two Gaussianrough paths on the first two levels.

Proposition 4.1.6. Let (X,Y ) = (X1, Y 1, · · · , Xd, Y d) : [0, T ]→ Rd+d be a centered Gaussianprocess with continuous sample paths where (Xi, Y i) and (Xj , Y j) are independent for i 6= j.Let ρ ∈ [1, 3

2) and assume that Vρ′(R(X,Y ), [0, T ]2) ≤ K < +∞ for a constant K > 0 where

ρ′ < ρ in the case ρ > 1 and ρ′ = 1 in the case ρ = 1. Let γ ≥ ρ such that 1γ + 1

ρ > 1.

Then there are constants C0, C1, C2 dependending on ρ, ρ′, γ and K and a control ω such thatω(0, T ) ≤ C0 and

|Xs,t − Ys,t|L2 ≤ C1 supu∈[s,t]

|Xu − Yu|1− ρ

γ

L2 ω(s, t)12γ

and ∣∣∣∣∫ t

sXs,u ⊗ dXu −

∫ t

sYs,u ⊗ dYu

∣∣∣∣L2

≤ C2 supu∈[s,t]

|Xu − Yu|1− ρ

γ

L2 ω(s, t)12γ

+ 12ρ

hold for every s < t.

Proof. Note first that, by assumption on Vρ′(R(X,Y ); [0, T ]2), Lemma 4.1.3 guarantees that thereis a control ω and a constant c1 = c1(ρ, ρ′) such that

Vρ(RX ; [s, t]2) ∨ Vρ(RY ; [s, t]2) ∨ Vρ(R(X−Y ); [s, t]2) ≤ ω(s, t)1/ρ

holds for all s < t and i = 1, . . . , d with the property that ω(0, T ) ≤ c1Kρ =: C0. We will

estimate both levels componentwise. We start with the first level. Let i ∈ 1, . . . , d. Then,∣∣Xis,t − Y i

s,t

∣∣2L2 =

∣∣∣∣R(Xi−Y i)

(s, ts, t

)∣∣∣∣≤ Vγ(R(X−Y ); [s, t]2)

95


and thus

|Xs,t − Ys,t|L2 ≤ c2

√Vγ(R(X−Y ); [s, t]2).

For the second level, consider first the case i = j. We have, using that (X,Y ) is Gaussian andthat we are dealing with geometric rough paths,∣∣∣∣∫ t

sXis,u dX

iu −

∫ t

sY is,u dY

iu

∣∣∣∣L2

=1

2

∣∣(Xis,t)

2 − (Y is,t)

2∣∣L2

=1

2

∣∣(Xis,t − Y i

s,t)(Xis,t + Y i

s,t)∣∣L2

≤ c3

∣∣Xis,t − Y i

s,t

∣∣L2

(|Xi

s,t|L2 + |Y is,t|L2

).

From the first part, we know that∣∣Xis,t − Y i

s,t

∣∣L2 ≤

√Vγ(R(X−Y ); [s, t]2).

Furthermore,

|Xis,t|L2 =

√∣∣∣∣RX ( s, ts, t

)∣∣∣∣ ≤√Vρ(RX ; [s, t]2) ≤ ω(s, t)12ρ

and the same holds for |Y is,t|L2 . Hence∣∣∣∣∫ t

sXis,u dX

iu −

∫ t

sY is,u dY

iu

∣∣∣∣L2

≤ c4

√Vγ(R(X−Y ); [s, t]2)ω(s, t)

12ρ .

For i 6= j, ∣∣∣∣∫ t

sXis,u dX

ju −

∫ t

sY is,u dY

ju

∣∣∣∣L2

≤∣∣∣∣∫ t

s(Xi − Y i)s,u dX

ju

∣∣∣∣L2

+

∣∣∣∣∫ t

sY is,u d(Xj − Y j)u

∣∣∣∣L2

.

We estimate the first term. From independence,

E

[(∫ t

s(Xi − Y i)s,u dX

ju

)2]

=

∫[s,t]2

R(Xi−Y i)

(s, us, v

)dRXj (u, v)

where the integral on the right is a 2D Young integral.1 By a 2D Young estimate (cf. [Tow02]),∣∣∣∣∣∫

[s,t]2R(Xi−Y i)

(s, us, v

)dRXj (u, v)

∣∣∣∣∣ ≤ c5(ρ, γ)Vγ(R(Xi−Y i); [s, t]2)Vρ(RXj ; [s, t]2)

≤ c6Vγ(R(X−Y ); [s, t]2)ω(s, t)1/ρ.

The second term is treated exactly in the same way. Summarizing, we have shown that∣∣∣∣∫ t

sXs,u ⊗ dXu −

∫ t

sYs,u ⊗ dYu

∣∣∣∣L2

≤ C√Vγ(R(X−Y ); [s, t]2)ω(s, t)

12ρ .

1The reader might feel a bit uncomfortable at this point asking why it is allowed to put expectation insidethe integral (which is not even an integral in Riemann-Stieltjes sense). However, this can be made rigorous bydealing with processes which have sample paths of bounded variation first and passing to the limit afterwards (cf.[FV10a, FV10b, FR, FH]). We decided not to go too much into detail here in order not to distract the readerfrom the main ideas and to improve the readability.

96


Finally, by Lemma 4.1.2

Vγ(R(X−Y ); [s, t]2) ≤ V∞(R(X−Y ); [s, t]2)1−ρ/γω(s, t)1/γ

and by the Cauchy-Schwarz inequality

V∞(R(X−Y ); [s, t]2) ≤ 4 supu∈[s,t]

|Xu − Yu|2L2


Corollary 4.1.7. Under the assumptions of Proposition 4.1.6, for every γ satisfying γ ≥ ρ and1γ + 1

ρ > 1, and every p > 2ρ and γ′ > γ, there is a (random) control ω such that

|Xns,t| ≤ ω(s, t)n/p (4.4)

|Yns,t| ≤ ω(s, t)n/p (4.5)

|Xns,t −Yn

s,t| ≤ εω(s, t)1

2γ′+n−1p (4.6)

holds a.s. for all s < t and n = 1, 2 where ε = supu∈[0,T ] |Xu − Yu|1− ρ

γ

L2 . Furthermore, there is aconstant C = C(p, ρ, γ, γ′,K) such that

|ω(0, T )|Lq ≤ CT (qp/2 + qγ′)

holds for all q ≥ 1.

Proof. Let ω be the control from Proposition 4.1.6. We know that

|Xns,t −Yn

s,t|L2 ≤ c1εω(s, t)12γ

+n−12ρ

holds for a constant c1 for all s < t and n = 1, 2. Furthermore, |Xns,t|L2 ≤ c2ω(s, t)

n2ρ for a

constant c2 for all s < t and n = 1, 2 and the same holds for Y (this just follows from settingY = const. and γ = ρ in Proposition 4.1.6). Now introduce a new process X : [0, T ] → Rd onthe same sample space as X such that for all sample points, we have

Xω(0,t)/ω(0,T ) = Xt, ∀t ∈ [0, T ],

and define Y in the same way. Then X, Y are well defined, multiplicative, and we can replacethe control ω by c3K|t−s| for the two re-parametrized processes. Using that X,Y are Gaussian,we may pass from L2 to Lq estimates and we know that O(|Xn|Lq) = qn/2 (same for Y andX−Y, cf. [FV10b, Appendix A]). Hence

|Xns,t|Lq ≤ c4(

√qK1/ρ)n|t− s|

n2ρ (4.7)

|Yns,t|Lq ≤ c4(

√qK1/ρ)n|t− s|

n2ρ (4.8)

|Xns,t − Yn

s,t|Lq ≤ εc4(

√qK1/ρ)n|t− s|

12γ

+(n−1)

2ρ (4.9)

hold for all s < t, n = 1, 2 and q ≥ 1 with ε = εK12γ− 1

2ρ . Using Lemma A.1.1 in the appendix,we see that there is a constant c5 = c5(p, ρ, γ, γ′,K) such that∣∣∣∣∣ sup

s<t∈[0,T ]

|Xns,t|

|t− s|n/p

∣∣∣∣∣Lq

≤ c5qn/2 (4.10)∣∣∣∣∣ sup

s<t∈[0,T ]

|Yns,t|

|t− s|n/p

∣∣∣∣∣Lq

≤ c5qn/2 (4.11)∣∣∣∣∣ sup

s<t∈[0,T ]

|Xns,t − Yn

s,t||t− s|1/p(n)

∣∣∣∣∣Lq

≤ εc5qn/2 (4.12)

97


hold for q sufficiently large and n = 1, 2 where 1p(n) = 1

2γ′ + n−1p . Set

ωnX(s, t) := supD⊂[s,t]

∑ti∈D|Xn

ti,ti+1|p/n

ωnY (s, t) := supD⊂[s,t]

∑ti∈D|Yn

ti,ti+1|p/n

ωnX−Y (s, t) := supD⊂[s,t]

∑ti∈D|Xn

ti,ti+1−Yn

ti,ti+1|p(n)

and

ω(s, t) :=∑n=1,2

ωnX(s, t) + ωnY (s, t) + ε1

p(n) ωnX−Y (s, t).

for s < t. Clearly, ω fulfils (4.4), (4.5) and (4.6). Moreover, the notion of p-variation is invariantunder reparametrization, hence

ωnX(0, T ) = supD⊂[0,T ]

∑ti∈D|Xn

ti,ti+1|p/n = sup

D⊂[0,T ]

∑ti∈D|Xn

ti,ti+1|p/n ≤ T sup

s<t∈[0,T ]

|Xns,t|p/n

|t− s|

and a similar estimate holds for ωnY (0, T ) and ωnX−Y (0, T ). By the triangle inequality and theestimates (4.7), (4.8) and (4.9),

|ω(0, T )|Lq ≤∑n=1,2

|ωnX(0, T )|Lq + |ωnY (0, T )|Lq + ε1

p(n) |ωnX−Y (0, T )|Lq

≤ c6T(qp/2 + q

p(1)2 + qp(2)

)≤ c7T (qp/2 + qγ

′)

for q large enough. We can extend the estimate to all q ≥ 1 by making the constant larger ifnecessary.

Corollary 4.1.8. Let ω be the random control defined in the previous corollary. Then, forevery n, there exists a constant cn such that

|Xns,t| < cnω(s, t)

np , |Yn

s,t| < cnω(s, t)np

a.s. for all s < t. The constants cn are deterministic and can be chosen such that cn ≤ 2n

(n/p)! ,

where x! := Γ(x− 1).

Proof. Follows from the extension theorem, cf. [Lyo98, Theorem 2.2.1] or [LCL07, Theorem3.7].

4.2 Main estimates

In what follows, we let p ∈ (2ρ, 3). Let γ ≥ ρ such that 1γ + 1

ρ > 1. We write log+ x = maxx, 0,and set

ε = supu∈[0,T ]

|Xu − Yu|1− ρ

γ

L2

98

Main estimates

2.1 Higher level estimates

We first introduce some notations. Suppose X is a multiplicative functional in TN (Rd) withfinite p-variation controlled by ω, N ≥ bpc. Then, define

Xs,t = 1 +

N∑n=1

Xns,t ∈ TN+1(Rd).

Then, X is multiplicative in TN , but in general not in TN+1. For any partition D = s = u0 <u1 < · · · < uL < uL+1 = t, define

XDs,t := Xs,u1 ⊗ · · · ⊗ XuL,t ∈ T

N+1(Rd).

The following lemma gives a construction of the unique multiplicative extension of X to higherdegrees. It was first proved in Theorem 2.2.1 in [Lyo98].

Lemma 4.2.1. Let X be a multiplicative functional in TN . Let D = s < u1 < · · · < uL < tbe any partition of (s, t), and Dj denote the partition with the point uj removed from D. Then,

XDs,t − XDj

s,t =

N∑n=1

Xnuj−1,uj ⊗XN+1−n

uj ,uj+1∈ TN+1(Rd). (4.13)

In particular, its projection onto the subspace TN is the 0-vector. Suppose further that X hasfinite p-variation controlled by ω, and N ≥ bpc, then the limit

lim|D|→0

XDs,t ∈ TN+1(Rd)

exists. Furthermore, it is the unique multiplicative extension of X to TN+1 with finite p-variation controlled by ω.

Theorem 4.2.2. Let (X,Y ) and ρ, γ as in Proposition 4.1.6. Then for every p > 2ρ and γ′ > γthere exists a constant C3 depending on p and γ′ and a (random) control ω such that for allq ≥ 1, we have

|ω(0, T )|Lq ≤M < +∞,

where M = M(p, ρ, γ, γ′,K, q), and the following holds a.s. for all [s, t]:

(i) If 12γ′ + 2

p > 1, then

|X3s,t −Y3

s,t| < C3εω(s, t)1

2γ′+2p .

(ii) If 12γ′ + 2

p = 1, then

|X3s,t −Y3

s,t| < C3ε · (1 + log+[ω(0, T )/ε

1− p2γ′]) · ω(s, t).

(iii) If 12γ′ + 2

p < 1, then

|X3s,t −Y3

s,t| < C3ε3−p

1−p/2γ′ ω(s, t),

99


Proof. Let s < t ∈ [0, T ] and let ω be the (random) control defined in Corollary 4.1.7. Then,by the same corollary, for every q ≥ 1, |ω(0, T )|Lq ≤ M . Fix an enhanced sample rough path(X,Y) up to level 2 and for simplicity, we will use ω to denote the corresponding realisation ofthe (random) control ω. We can assume without loss of generality that

ε < ω(s, t)1p− 1

2γ′ , (4.14)

otherwise there will be nothing to prove. Let D = s = u0 < · · · < uL+1 = t be a dissection.Then (cf. [Lyo98, Lemma 2.2.1]), there exists a j such that

ω(uj−1, uj+1) ≤ 2

Lω(s, t), L ≥ 1. (4.15)

Let Dj denote the dissection with the point uj removed from D. Then, we have

|(XDs,t − YD

s,t)3| < |(XDj

s,t − YDj

s,t )3|+2∑

k=1

(|Rkuj−1,uj ⊗X3−k

uj ,uj+1|

+ |Xkuj−1,uj ⊗R3−k

uj ,uj+1|+ |Rk

uj−1,uj ⊗R3−kuj ,uj+1

|),

where Rs,t = Ys,t −Xs,t. By assumption,

|Rkuj−1,uj ⊗R3−k

uj ,uj+1| < C ·min

ε( 1

Lω(s, t)

) 12γ′+

2p ,( 1

Lω(s, t)

) 3p

, (4.16)

and similar inequalities hold for the other two terms in the bracket. Thus, we have

|(XDs,t − YD

s,t)3| < |(XDj

s,t − YDj

s,t )3|+ C3 min

ε( 1

Lω(s, t)

) 12γ′+

2p ,( 1

Lω(s, t)

) 3p

.

Let N be the integer that

[1

N + 1ω(s, t)]

1p− 1

2γ′ ≤ ε < [1

Nω(s, t)]

1p− 1

2γ′ , (4.17)

then

ε[1

Lω(s, t)]

12γ′+

2p < [

1

Lω(s, t)]

3p

if and only if L ≤ N . By Lemma 4.2.1, we have

X3s,t = lim

|D|→0(XD

s,t)3, Y3

s,t = lim|D|→0

(YDs,t)

3.

Thus, for a fixed partition D, we choose a point each time according to (4.15), and drop themsuccessively. By letting |D| → +∞, we have

|X3s,t −Y3

s,t| ≤ C3

[εN∑L=1

( 1

Lω(s, t)

) 12γ′+

2p +

+∞∑L=N+1

( 1

Lω(s, t)

) 3p

].

Approximating the sums by integrals, we have

|X3s,t −Y3

s,t| < C3[εω(s, t)1

2γ′+2p (1 +

∫ N

1x−( 1

2γ′+2p

)dx) + ω(s, t)

3p

∫ +∞

Nx− 3pdx].

Compute the second integral, and use

[1

N + 1ω(s, t)]

( 1p− 1

2γ′ ) ≤ ε,

100

Main estimates

we obtain

|X3s,t −Y3

s,t| < C3[εω(s, t)1

2γ′+2p (1 +

∫ N

1x−( 1

2γ′+2p

)dx) + ε

3−p1−p/2γ′ ω(s, t)]. (4.18)

Now we apply the above estimates to the three situations respectively.

1. 12γ′ + 2

p > 1.

In this case, the integral ∫ N

1x−( 1

2γ′+2p

)dx <

∫ +∞

1x−( 1

2γ′+2p

)dx < +∞

converges. On the other hand, (4.14) implies

ε3−p

1−p/2γ′ ω(s, t) < εω(s, t)1

2γ′+2p ,

thus, from (4.18), we get

|X3s,t −Y3

s,t| < C3εω(s, t)1

2γ′+2p .

2. 12γ′ + 2

p = 1.

In this case, 1p −

12γ′ = 3−p

p , and 3−p1−p/2γ′ = 1. Thus, by the second inequality in (4.17), we have∫ N

1x−1dx = logN < logω(s, t)− p

3− plog ε.

On the other hand, (4.14) gives

logω(s, t)− p

3− plog ε > 0.

Combining the previous two bounds with (4.18), we get

|X3s,t −Y3

s,t| < C3ε[1 + logω(s, t)− p

3− plog ε]ω(s, t).

We can simplify the above inequality to

|X3s,t −Y3

s,t| < C3ε[1 + log+(ω(0, T )/εp

3−p )]ω(s, t),

where we have also included the possibility of ε ≥ ω(0, T )3p−1

.

3. 12γ′ + 2

p < 1.

Now we have

1 +

∫ N

1x−( 1

2γ′+2p

)dx < CN

1− 12γ′−

2p < C · ε−(1− 1

2γ′−2p

)/ 1p− 1

2γ′ ω(s, t)1− 1

2γ′−2p ,

where the second inequality follows from (4.17). Combining the above bound with (4.18), weobtain

|X3s,t −Y3

s,t| < C3ε3−p

1−p/2γ′ ω(s, t).

101


The following theorem, obtained with the standard induction argument, gives estimates forall levels n = 1, 2, · · · .

Theorem 4.2.3. Let (X,Y ) and ρ, γ as in Proposition 4.1.6, p > 2ρ and γ′ > γ. Then thereexists a (random) control ω such that for every q ≥ 1, we have

|ω(0, T )|Lq ≤M

where M = M(p, ρ, γ, γ′, q,K), and for each n there exists a (deterministic) constant Cn de-pending on p and γ′ such that a.s. for all [s, t]:

(i) If 12γ′ + 2

p > 1, then we have

|Xns,t −Yn

s,t| < Cnεω(s, t)1

2γ′+n−1p

(ii) If 12γ′ + 2

p = 1, then we have

|Xns,t −Yn

s,t| < Cnε · (1 + log+[ω(0, T )/ε

1− p2γ′]) · ω(s, t)

12γ′+

n−1p .

(iii) If 12γ′ + 2

p < 1, then for all s < t and all small ε, we have

|Xns,t −Yn

s,t| < Cnε3−p

1−p/2γ′ ω(s, t)n−1+p

p . (4.19)

Proof. We prove the case when 12γ′ + 2

p < 1; the other two situations are similar. Let ω bethe control in the previous theorem. Fix an enhanced sample path (X,Y), the correspondingrealisation ω of ω, and s < t ∈ [0, T ]. We may still assume (4.14) without loss of generality.Thus, for n = 1, 2, we have

|Xns,t −Yn

s,t| < εω(s, t)1

2γ′+n−1p < Cnε

3−p1−p/2γ′ ω(s, t)

n−1+pp ,

where the second inequality comes from (4.14). The above inequality also holds for k = 3 bythe previous theorem. Now, suppose (4.19) holds for k = 1, · · · , n, where n ≥ 3, then for levelk = n+ 1, the exponent is expected to be

n+ pp

> 1,

so that the usual induction procedure works (cf. [Lyo98], Theorem 2.2.2.). Thus, we prove(4.19) for all n.

2.2 Proof of Theorem 4.0.1

Proof. We prove the second situation when 12γ + 1

ρ ≤ 1. The first one is similar. Let ε =

supu∈[0,T ] |Xu−Yu|1− ρ

γ

L2 . It is sufficient to show that for every p > 2ρ there is a constant C suchthat

|ρNσ−var(X,Y)|Lq ≤ Cε3−p

1−ρ/γ ,

where σ > 2γ and N ≥ bσc both satisfy the assumptions of Theorem 4.0.1. Set

ρ′ := (1 + η)ρ, p := 2(1 + 2η)ρ, γ′ := (1 + η)γ, γ′′ := (1 + 2η)γ

102

Applications

for some η > 0. We can choose η small enough such that 1ρ′ + 1

γ′ > 1 and p < 3 hold, and

the conditions of Theorem 4.2.3 are satisfied for ρ′ and γ′. Clearly 1γ′′ + 2

p <1

2γ + 2p ≤ 1, thus

Theorem 4.2.3 implies that

|Xns,t −Yn

s,t| < Cnε3−p

1−ρ/γ ω(s, t)n−1+p

p

holds a.s. for any n and s < t where ω is a random control as in Theorem 4.2.3. Furthermore,for any n,

n− 1 + pp

=n

2γ′′+ n(

1

p− 1

2γ′′)− 1− p

p=

n

2γ′′+ (n− 1)(

1

p− 1

2γ′′) + (1− 2

p− 1

2γ′′).

(4.20)

Note that the last expression implies that

θn := n(1

p− 1

2γ′′)− 1− p

p> 0

for all n. Fix a dissection D = 0 = u0 < . . . < uL < T of the interval [0, T ]. Usingω(ui, ui+1) ≤ ω(0, T ), we have(∑

i

|Xnui,ui+1

−Ynui,ui+1

|σn

)nσ

≤ Cnε3−p

1−ρ/γ ω(0, T )θn(∑

i

ω(ui, ui+1)σ

2γ′′

)nσ

.

Choosing η smaller if necessary, we may assume that σ ≥ 2γ′′ and super-additivity of the controlimplies (∑

i

ω(ui, ui+1)σ

2γ′′

)nσ

≤ ω(0, T )n

2γ′′ .

Passing to the supremum over all partitions of [0, T ], we have

supD

(∑i

|Xnui,ui+1

−Ynui,ui+1

|σn

)nσ

≤ Cnε3−p

1−ρ/γ ω(0, T )n−1+p

p .

Let q ≥ 1. By Theorem 4.2.3, there is a constant M depending on ρ, γ, σ, δ, q and K such that|ω(0, T )|Lq ≤M . Taking Lq norm on both sides, we have

|ρNσ−var(X,Y)|Lq ≤ Cε3−p

1−ρ/γ

which was the claim.

4.3 Applications

3.1 Convergence rates of rough differential equation

Consider the rough differential equation of the form

dYt =

d∑i=1

Vi(Yt) dXit =: V (Yt) dXt; Y0 ∈ Re (4.21)

103


where X is a centered Gaussian process in Rd with independent components and V = (Vi)di=1 a

collection of bounded, smooth vector fields with bounded derivatives in Re. Rough path theorygives meaning to the pathwise solution to (4.21) in the case when the covariance RX has finiteρ-variation for some ρ < 2. Assume that ρ ∈ [1, 3

2) and that there is a constant K such that

Vρ(RX ; [s, t]2) ≤ K|t− s|1ρ (4.22)

for all s < t (note that this condition implies that the sample paths of X are α-Holder for allα < 1

2ρ). For simplicity, we also assume that [0, T ] = [0, 1]. For every k ∈ N, we can approximatethe sample paths of X piecewise linear at the time points 0 < 1/k < 2/k < . . . < (k−1)/k < 1.We will denote this process by X(k). Clearly, X(k) → X uniformly as k →∞. Now we substituteX by X(k) in (4.21), solve the equation and obtain a solution Y (k); we call this the Wong-Zakaiapproximation of Y . One can show, using rough path theory, that Y (k) → Y a.s. in uniformtopology as k →∞. The proposition below is an immediate consequence of Theorem 4.0.1 andgives us rates of convergence.

Proposition 4.3.1. The mesh size 1k Wong-Zakai approximation converges uniformly to the

solution of (4.21) with a.s. rate at least k−( 3

2ρ−1−δ)

for any δ ∈ (0, 32ρ − 1). In particular, the

rate is arbitrarily close to 12 when ρ = 1, which is the sharp rate in that case.

Proof. First, one shows that (4.22) implies that

supt∈[0,1]

|X(k)t −Xt|L2 = O(k

− 12ρ ).

One can show (cf. [FV10b, Chapter 15.2.3]) that there is a constant C such that

supk∈N

Vρ(R(X,X(k)); [s, t]2) ≤ C|t− s|1ρ

holds for all s < t. By choosing q large enough, a Borel-Cantelli type argument applied toTheorem 4.0.1 shows that ρσ−var(X,X

(k))→ 0 a.s. for k →∞ with rate arbitrarily close to

k− 1

2( 1ρ− 1γ

), if

1

2γ+

1

ρ> 1,

and arbitrarily close to

k−(3−2ρ), if1

2γ+

1

ρ≤ 1,

both cases are subject to γ ≥ 32 and 1

γ + 1ρ > 1. Note that in the second situation, the actual

value of γ does not matter, and we always have a rate of ’almost’ 32ρ − 1. For the first situation,

we need to let γ as large as possible but still satisfy the constraints. The critical value is1γ∗ = 1

2(1 − 1ρ), which also results in a rate that is arbitrarily close to 3

2ρ − 1. Using the localLipschitz property of the Ito Lyons map (cf. [FV10b, Theorem 10.26]), we conclude that theWong-Zakai convergence rate is faster than

k−( 3

2ρ−1−δ)

for any δ > 0 (but not for δ = 0).

104

Applications

Remark 4.3.2. For ρ ∈ (1, 32), the rate above is not optimal. In fact, the sharp rate in this case

is ’almost’ 1ρ −

12 , as shown in [FR]. The reason for the non-optimality of the rate is that we

obtain the third level estimate merely based on the first two levels, which leads to a reduction inthe exponent in the rate. On the other hand, this method does not use any Gaussian structureon the third level, and can be applied to more general processes. For the case ρ = 1, we recoverthe sharp rate of ’almost’ 1

2 .

3.2 The stochastic heat equation

In the theory of stochastic partial differential equations (SPDEs), one typically considersthe SPDE as an evolution equation in a function space. When it comes to the question of timeand space regularity of the solution, one discovers that they will depend on the particular choiceof this space. As a rule of thumb, the smaller the space, the lower the time regularity ([Hai09],Section 5.1). The most prominent examples of such spaces are Hilbert spaces, typically Sobolevspaces. However, in some cases, it can be useful to choose rough paths spaces instead ([Hai11]).A natural question now is whether the known regularity results for Hilbert spaces are also truefor rough paths spaces. In this section, we study the example of a modified stochastic heatequation for which we can give a positive answer.

Consider the stochastic heat equation:

dψ = (∂xx − 1)ψ dt+ σ dW (4.23)

where σ is a positive constant, the spatial variable x takes values in [0, 2π], W is space-time whitenoise, i.e. a standard cylindrical Wiener process on L2([0, 2π],Rd), and ψ denotes the stationarysolution with values in Rd. The solution ψ is expected to be almost 1

4 -Holder continuous in timeand almost 1

2 -Holder continuous in space (cf. [Hai09]). In the next Theorem, we show that thisis indeed the case if we choose the appropriate rough paths space.

Theorem 4.3.3. Let p > 2. Then, for any fixed t ≥ 0, the process x 7→ ψt(x) is a Gaussianprocess (in space) which can be lifted to an enhanced Gaussian process Ψt(·), a process withsample paths in C0,p−var([0, 2π], Gbpc(Rd)). Moreover, t 7→ Ψt(·) has a Holder continuous modi-

fication (which we denote by the same symbol). More precisely, for every α ∈(

0, 14 −

12p

), there

exists a (random) constant C such that

ρp−var(Ψs,Ψt) ≤ C|t− s|α

holds almost surely for all s < t. In particular, choosing p large gives a time regularity of almost14 -Holder.

Proof. The fact that x 7→ ψt(x) can be lifted to a process with rough sample paths and thatthere is some Holder-continuity in time was shown in Lemma 3.1 in [Hai11], see also [FH]. Wequickly repeat the argument and show where we can use our results in order to derive the exactHolder exponents. Using the standard Fourier basis

ek(x) =

1√π

sin(kx) if k > 01√2π

if k = 01√π

cos(kx) if k < 0

the equation (4.23) can be rewritten as a system of SDEs

dY kt = −(k2 + 1)Y k

t dt+ σ dW kt

105


where (W k)k∈Z is a collection of independent standard Brownian motions and (Y k)k∈Z arethe stationary solutions of the SDEs, i.e. a collection of centered, independent, stationaryOrnstein-Uhlenbeck processes. The solution of (4.23) is thus given by the infinite sum ψt(x) =∑

k∈Z Ykt ek(x). One can easily see that

E [ψs(x)⊗ ψt(y)] =σ2

4π

∑k∈Z

cos(k(x− y))

1 + k2e−(1+k2)|t−s| × Id

where Id denotes the identity matrix in Rd×d. In particular, for s = t,

E [ψt(x)⊗ ψt(y)] = K(x− y)× Id

where K is given by

K(x) =σ2

4 sinh(π)cosh(|x| − π)

for x ∈ [−π, π] and extended periodically for the remaining values of x (this can be derived bya Fourier expansion of the function x 7→ cosh(|x| − π)). In particular, one can calculate thatx 7→ ψt(x) is a Gaussian process with covariance of finite 1-variation (see the remark at the endof the section for this fact), hence ψt can be lifted to process Ψt with sample paths in the roughpaths space C0,p−var([0, 2π], Gbpc(Rd)) for any p > 2.

Furthermore, for any s < t, x 7→ (ψs(x), ψt(x)) is a Gaussian process which fulfils theassumptions of Theorem 4.0.1 and the covariance R(ψs,ψt) also has finite 1-variation, uniformlybounded for all s < t, hence

sups<t|R(ψs,ψt)|1−var;[0,2π]2 =: c1 <∞.

Therefore, for any γ ∈ (1, p/2) and q ≥ 1 there is a constant C = C(p, γ, c1, q) such that

|ρp−var(Ψs,Ψt)|Lq ≤ C supx∈[0,2π]

|ψt(x)− ψs(x)|1− 1

γ

L2

holds for all s < t. A straightforward calculation (cf. [Hai11, Lemma 3.1]) shows that

|ψt(x)− ψs(x)|L2 ≤ c2|t− s|1/4

for a constant c2. In particular, we can find γ and q large enough such that

α <

q4(1− 1

γ )− 1

q=

(1

4− 1

4γ

)− 1

q<

1

4− 1

2p.

Since C0,p−var is a Polish space, we can apply the usual Kolmogorov continuity criterion toconclude.

Remark 4.3.4. We emphasize that here, for every fixed t, the process ψt(·) is a Gaussianprocess, where the spatial variable x should now be viewed as ’time’. This idea is due to M.Hairer.Knowing that the spatial regularity is ’almost’ 1/2 for every fixed time t, one could guess thatcovariance of this spatial Gaussian process has finite 1-variation. For a formal calculation, werefer to [Hai11] or [FH].

106

5

Spatial rough path lifts of stochasticconvolutions

The lack of spatial regularity of solutions to PDE subjected to space-time white noise (or otherinfinite-dimensional noise) can cause serious obstacles concerning well-posedness and stability.In recent work Hairer and coauthors [Hai11, HW13, Hai, HMW13] realized that several inter-esting, at first sight ill-posed, non-linear stochastic PDE can be solved by constructing a spatialrough path associated to the linearized equation. This program was carried out for a system ofstochastic Burger’s equations (with motivation from path sampling problems) and more recentlyfor the KPZ equation on the one dimensional torus. In both cases, the linearized equation isthe classical one dimensional stochastic heat equation

dΨt = (∆− 1) Ψtdt+ dWt, on [0, 2π], (SHE)

with periodic boundary conditions. Here ∆ is the one dimensional Laplacian, i.e. ∆ = ∂2x,

the summand −1 allows to consider the corresponding stationary solution and W is space-timewhite noise integrated in time. We shall follow Hairer’s setup and consider d i.i.d. realizations of(SHE). The question then arises how to construct a spatial rough path over x 7→ Ψ (t, x;ω) forfixed time t. If such a rough path lift has been constructed one can view (SHE) as an evolutionin a rough path space, a point of view which has proven extremely fruitful in solving newclasses of until now ill-posed stochastic PDE. In [Hai11] Hairer succeeded in establishing finite1-variation (in 2D) sense of the covariance of the stationary solution to (SHE), i.e. of (x, y) 7→EΨ (t, x) Ψ (t, y). According to (Gaussian) rough path theory in the form of [FV10a, FV10b]this gives a “canonical” way of lifting Ψ to a rough path Ψ. The rough path Ψ is a “level 2”rough path or more precisely: Ψ is a 1/p-Holder geometric rough path for any p > 2 as functionof x, with local regularity properties akin to standard Brownian motion as function of t.

It is clear that the Brownian-like regularity of x 7→ Ψ (t, x;ω) is due to the competitionbetween the smoothing effects of the Laplacian and the roughness of space-time white noise.Truncation of the higher noise modes (or suitable “coloring”) leads to better spatial regularity;on the other hand, replacing ∆ by a fractional Laplacian, i.e. considering

dΨt = (− (−∆)α − 1) Ψtdt+ dWt, on [0, 2π], (fSHE)

for some α ∈ (0, 1) dampens the smoothing effect and x 7→ Ψ (t, x;ω) will have “rougher”regularity properties than a standard Brownian motion. One thus expects ρ-variation regularityfor the spatial covariance of x 7→ Ψ (t, x;ω) only for some ρ > 1 and subsequently only theexistence of a “rougher” rough path, i.e. necessarily with higher p than before. Applied in thepresent context, our main insights/results are as follows:

• The local covariance/decorrelation structure of the stationary fractional stochastic heatequation (fSHE) (in x) is akin to the same structure for fractional Brownian motion (in

Rough path lifts of stoch. convolutions

t). In quantitative terms,

fBM with Hurst parameter H ↔ fSHE with α =1

2+H.

As a consequence one has finite 2D-ρ-variation of the covariance1 with

ρ =1

2Hfor fBM with H ≤ 1

2,

ρ =1

2α− 1for fSHE with α ≤ 1.

• Recalling criticality of ρ∗ = 2 we are able to lift the stationary solution to (fSHE) in thespatial variable to a rough path provided

α > α∗ =3

4,

similar to the well-known condition H > H∗ = 14 for fBM (cf. [CQ02]). More precisely,

the resulting (geometric rough) path enjoys 1p -Holder regularity for any p > 2ρ = 2

2α−1 .

When α > 56 we have ρ = 1

2α−1 <32 and can pick p < 3. The resulting rough path can

then be realized as a “level 2” rough path. In the general case (similar to H ∈ (14 ,

13 ] in the

fBM setting) one must go beyond the stochastic area and control the third level iteratedintegrals. We emphasize that the notoriously difficult third-level computation need notbe repeated in the present context. Everything is obtained as application of availablegeneral theory, once finite ρ-variation of the covariance is established. A satisfactoryapproximation theory is also available, based on uniform ρ-variation estimates.

• On a more technical level, we give a novel criterion to control the 2D-variation of thecovariance of (not-necessarily Gaussian) processes with stationary increments and variancesatisfying a decay condition at 0 and a concavity property; this translates into the requiredlocal covariance/decorrelation structure to which we alluded in the first point above.Checking this criterion for fBM is essentially trivial. For fSHE this involves a fair amountof Fourier analysis, as does the quest for uniform 2D-ρ-variation estimates, e.g. in thecase of smoothing fSHE with hyper-viscosity. Among others, we are then led to considerconvexity, Holder regularity and L1-estimates for Fourier series.

We believe that following a similar route as Hairer in [Hai]2 our techniques may prove usefulin solving the fractional KPZ equation

dXt = −(−∆)αXtdt+ (∇Xt)2dt+ dWt, on R,

with α ∈ (0, 1). Such fractional KPZ equations arise as models of growing surfaces with impu-rities (cf. [MW97, XTHX12, Kat03]).

Another motivation are path sampling problems for diffusions driven by fractional noise.Recall that the original motivation for studying vector-valued stochastic Burger’s equationsemerged in path sampling problems for SDE of the form

dZu = AZudu+ f(Zu)du+ CdBu, in Rd, (5.1)

on [0, 2π]. Here f : Rd → Rd is a possibly non-linear function and B denotes standard Brownianmotion in Rd. In a series of papers [HSVW05, HSV07, Hai11] the authors realized that the law

1The result for fBM is of course known (cf. [FV11] and the references therein).2Let us note that Hairer considers the KPZ equation on the torus while most literature is on the whole real

line. In the end of Section 5.3 we demonstrate that our results may be used to construct local spatial rough pathlifts of solutions to (fSHE) on the real line.

108

Main Result

L(Z) on C([0, 2π];Rd) and conditioned on the endpoints coincides with the invariant measureof vector-valued stochastic Burger’s equations of the type

dXit = ∆Xi

t +n∑j=1

gij(Xt)∂xXjt dt+ dW i

t , i = 1, ..., d, (5.2)

with appropriate boundary conditions. Based on this, efficient sampling algorithms for thepaths of Z conditioned on the endpoints may be derived. It is tempting to try a similarapproach in the fractional case; that is, to sample the law of (5.1) with B replaced by fractionalBrownian motion BH conditional on its endpoints, via the stationary solution of a suitablefractional SPDE. However, combining the heuristics found in [HSV07] suggests an SPDE of theform (fSHE) with appropriate boundary conditions and with an additional non-local, nonlinearterm. At present, handling the resulting SPDE is an open problem, although one suspects thatthe present considerations will prove useful in this regard.

As this work near completion we learned that Gubinelli and coauthors [GIP12] adaptedHairer’s analysis of the stochastic Burger’s equation to the fractional case (i.e. (5.2) with ∆replaced by −(−∆)α for α > 5

6) and established existence and uniqueness of solutions. In doingso they construct a rough path associated to fSHE by direct calculations under the strongercondition α > 5

6 . In contrast, our construction of the associated rough path is based on generalGaussian rough path theory and thus allows immediate application of the corresponding generalresults, such as stability under approximations and moment estimates. Without further effort wededuce estimates on the rate of convergence of approximations following from general Gaussianrough path techniques. On the other hand, a very important contribution of [GIP12] is thepossibility of going beyond space dimension one which is not at all within the scope of ourconsiderations here.

The chapter is structured as follows: In Section 5.1 we establish a general condition provingfinite ρ-variation for processes with stationary increments and convex or concave variance func-tion. We then use this condition to prove the existence of rough path lifts for such processes.The application of this general result to stationary solutions of fractional stochastic heat equa-tions thus requires proof of concavity of their (spatial) variance function. Sufficient conditionsfor the concavity of such variance functions in terms of their Fourier coefficients are derivedin Section 5.2 and used in Section 5.3 to lift strictly stationary Ornstein-Uhlenbeck processescorresponding to fractional stochastic heat equations to Gaussian rough paths with respect totheir space variable. Moreover, we prove strong convergence in rough path metric of hyper-viscosity and Galerkin approximations. Section 5.3 is then concluded by a brief investigation ofthe stochastic fractional heat equation on the whole real line.

5.1 Main Result

In this section, we shall consider centered, continuous stochastic processes with IID compo-nents X =

(X1, . . . , Xd

)and stationary increments. If X is Gaussian, the construction of a

(geometric) rough path associated to X then naturally passes through an understanding of thetwo-dimensional ρ-variation of the covariance of X1. For brevity, we abuse notation and writeX ≡ X1 until further notice. The law of such a process is fully determined by

σ2 (u) := EX2t,t+u = RX

(t, t+ ut, t+ u

).

Lemma 5.1.1. (i) Assume that σ2 (·) is concave on [0, h] for some h > 0. Then, non-overlapping increments are non-positively correlated in the sense that

EXs,tXu,v = RX

(s, tu, v

)≤ 0,

109


for all 0 ≤ s ≤ t ≤ u ≤ v ≤ h.

(ii) Assume in addition that σ2 (·) restricted to [0, h] is non-decreasing. Then,

0 ≤ EXs,tXu,v = |EXs,tXu,v| ≤ EX2u,v = σ2 (v − u) ,

for all 0 ≤ s ≤ u ≤ v ≤ t ≤ h.

Proof. (1): This result can be found in [MR06, Lemma 7.2.7]. It just follows from the identity

2EXs,tXu,v = σ2(v − s) + σ2(u− t)− σ2(v − t)− σ2(u− s)

and the concavity of the function σ2.(2): Note Xs,tXu,v = (a+ b+ c) b where a = Xs,u, b = Xu,v, c = Xv,t. Applying the

algebraic identity2 (a+ b+ c) b = (a+ b)2 − a2 + (c+ b)2 − c2

and taking expectations yields

2EXs,tXu,v = EX2s,v − EX2

s,u +X2u,t − EX2

v,t

=(σ2 (v − s)− σ2 (u− s)

)+(σ2 (t− u)− σ2 (t− v)

)≥ 0 + 0,

where we used that σ2 (·) is non-decreasing. We thus see

0 ≤ EXs,tXu,v = |EXs,tXu,v| .

On the other hand, from (a+ b+ c) b = b2 + ab + cb, and then non-positive correlation of thenon-overlapping increments,

EXs,tXu,v = EX2u,v + EXs,uXu,v + EXv,tXu,v︸︷︷︸

≤0

≤ EX2u,v.

This concludes the proof of part (ii).

Theorem 5.1.2. Let X be a real-valued stochastic process on [0, T ] with stationary incrementsand covariance function σ2(u) := E(Xu−X0)2. Assume that σ2 : [0, T ]→ R+ is concave and that

|σ2(u)| ≤ Cσ|u|1ρ for all u ∈ [0, T ]. Assume further that for some h > 0, σ2

|[0,h] is non-decreasing.

Then the covariance of X is of finite ρ-variation on every rectangle [s, t]× [u, v] ⊆ [0, T ]2.More precisely, if the interior of [s, t] × [u, v] does not intersect with the diagonal D =

(x, x) : x ∈ [0, T ], then

Vρ(RX ; [s, t]× [u, v])ρ ≤ Cρσ√|t− s|

√|v − u|. (5.3)

If [s, t]× [u, v] is contained in the strip Sh =

(x, y) ∈ [0, 1]2 : |x− y| ≤ h

, then

Vρ(RX ; [s, t]× [u, v])ρ ≤ C(|t− s| ∧ |v − u|) (5.4)

holds for some constant C = C(ρ, Cσ) > 0.In particular, there is a constant C1 = C1(ρ, Cσ, h) such that

Vρ(RX ; [s, t]2)ρ ≤ C1|t− s|

holds for all [s, t]2 ⊆ [0, T ]2 and the covariance of X has finite Holder-controlled ρ′-variationfor all ρ′ > ρ on [0, T ].

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Using these estimates in (5.5) yields∑tj

|EXti,ti+1Xt′j ,t′j+1|ρ ≤ C3|ti+1 − ti|

for some constant C3 and we conclude as in case 1.

Case 2: We use the estimates from the cases 1 and 3 to see that

21−ρVρ(RX ; [s, t]× [u, v])ρ ≤ Vρ(RX ; [s, t]× [u, s])ρ + Vρ(RX ; [s, t]× [s, v])ρ

≤ C1|s− u|+ C3|v − s|≤ (C1 ∨ C3)|v − u|.

This finishes the proof of (5.4)

If [s, t] × [u, v] is any rectangle contained in [0, T ]2, it can be written as a finite union ofrectangles A1, . . . , An which are contained in Sh or lie off-diagonal. From the inequality

Vρ(RX ; [s, t]× [u, v]) ≤ C(ρ, n)n∑k=1

Vρ(RX ;Ak)

it follows that indeed the ρ-variation is is finite over all rectangles.

From equation (5.4), we know that

Vρ(RX ; [s, t]2)ρ ≤ C|t− s|

holds for all [s, t]2 provided that |t − s| ≤ h. Using (5.3) and choosing the constant C larger,we may conclude that the estimate holds for all squares [s, t]2 ⊆ [0, T ]2. With [FV11, Theorem1] this implies the claimed finite Holder-controlled ρ′-variation for all ρ′ > ρ.

In the sense of the following Remark, the assumption of σ2 to be non-decreasing on [0, h]in Theorem 5.1.2 is superfluous. We have required it only in order to get a control on theρ-variation on the complete interval [0, h].

Remark 5.1.3. Let σ2 : [0, T ] → R+ be a continuous, concave function with σ2(0) = 0 forsome T > 0. Then there is an h ∈ (0, T ] such that σ2 is non-decreasing on [0, h].

Proof. Since σ2 : [0, T ]→ R+ is a continuous, concave function, each local maximum is a globalmaximum. Let h := inf

t ∈ [0, T ]| σ2(t) = maxr∈[0,T ] σ

2(r)

. Without loss of generality wemay assume σ2 6≡ 0. Thus, h > 0 and σ2 is non-decreasing on [0, h], since otherwise σ2 wouldhave a local maximum in [0, h), thus attaining maxr∈[0,T ] σ

2(r) in [0, h) in contradiction to thedefinition of h.

A continuous, centered Gaussian process satisfying the assumptions from Theorem 5.1.2with ρ ∈ [1, 2) can be lifted to a geometric p-rough path on the interval [0, T ] (cf. [FV10a,Theorem 35], [FV10b, Theorem 15.33]). Thus, we obtain

Corollary 5.1.4. Let X = (X1, ..., Xd) : [0, T ] → Rd be a centered continuous stochastic pro-cess with independent components such that each Xi has stationary increments with covariancefunction σ2,i(u) := E(Xi

u−Xi0)2. Assume that for all i = 1, . . . , d, σ2,i : [0, T ]→ R+ is concave,

|σ2,i(u)| ≤ Cσ|u|1ρ for all u ∈ [0, T ] and some ρ ≥ 1 and that there is an h > 0 such that σ2

|[0,h]

is non-decreasing. Then the covariance function R : [0, T ]2 → Rd×d has finite ρ-variation andfinite Holder-controlled ρ′-variation for every ρ′ > ρ. In particular, if X is Gaussian and ρ < 2,there exists a continuous G[p](Rd)-valued process X such that

112

Conditions in terms of Fourier coefficients

(i) X is a geometric p-rough path with X ∈ C0, 1p−Holder

0 ([0, T ], G[p](Rd)) almost surely forevery p > 2ρ,

(ii) X lifts X in the sense that π1(Xt) = Xt −X0,

(iii) There is a C = C(ρ, Cσ, h) such that for all s < t in [0, T ] and q ∈ [1,∞),

|d(Xs,Xt)|Lq ≤ C√q|t− s|

12ρ

(iv) (Fernique-estimates) for all p > 2ρ there exists η = η(p, ρ, h) > 0, such that

Eeη‖X‖21

p−Hol;[0,T ] <∞.

Remark 5.1.5. At this point, one might ask what can be said in the case where σ2(u) =E(Xu−X0)2 is convex on the interval [0, h]. Proceeding as in Lemma 5.1.1 (i), it can be shownthat non-overlapping increments are non-negatively correlated in this case. By writing largeincrements Xs,t as a sum of smaller increments Xs,t = Xs,u1 +Xu1,u2 +Xu2,t, s ≤ u1 ≤ u2 ≤ t,it then follows that

RX

(s, tu, v

)= EXs,tXu,v ≥ 0,

for every rectangle [s, t]× [u, v] ⊆ [0, h]2. In other words, every rectangular increment of RX ispositive. This readily implies finite 1-variation over every rectangle [s, t]× [u, v] ⊆ [0, h]2. Moreprecisely, we have the estimate

V1(RX ; [s, t]× [u, v]) ≤ |EXs,tXu,v|.

Furthermore, by convexity, 0 ≤ σ2(u) ≤ uσ2(1) for u ∈ [0, 1] we deduce

V1(RX ; [s, t]× [u, v]) .√|t− s|

√|v − u|,

for all [s, t]2 ⊆ [0, h]2.The same is of course true for multidimensional processes X = (X1, . . . , Xd) under the

condition that every σ2;i(u) = E(Xiu−Xi

0)2 is convex on [0, h]. if X is Gaussian, this implies that

we can lift X to a process with sample paths in the rough paths space C0, 1p−Holder

0 ([0, h], G[p](Rd))for every p > 2. Moreover, the concatenation of geometric rough paths over adjacent intervals isagain a rough path (cf. [CLL12], Lemma 4.9) and hence we can lift the sample paths of Gaussianprocesses X : [0, T ]→ Rd to Gaussian rough paths on the whole interval [0, T ] provided there isa (possibly small) h > 0 such that every σ2;i is convex on [0, h].

For instance, this applies to fractional Brownian motion with Hurst parameter H ≥ 1/2.However, from a rough paths point of view, these cases are rather trivial since we can use Itocalculus or Young’s integration theory to define iterated integrals.

5.2 Conditions in terms of Fourier coefficients

The application of Theorem 5.1.2 to stationary solutions Ψ(x) to the stochastic fractional heatequation (fSHE), as functions of their spatial variable, leads to continuous, stationary, centeredGaussian processes Ψ : S1 → R, S1 = [0, 2π]/0, 2π, with Fourier decomposition

Ψ(x) =∑k∈Z

Zkeikx,

113


where Zk are complex Gaussian random variables3 with covariance

EZkZ l = δk,lak2

and real valued coefficients ak satisfying ak = a−k for all k ∈ N. Then, the covariance R is ofthe form

RK(x, y) = EΨ(x)Ψ(y) = EΨ(x)Ψ(y) =1

2

∑k∈Z

akeik(x−y), ∀x, y ∈ [0, 2π].

Therefore,

RK(x, y) = K(|x− y|) =a0

2+∞∑k=1

ak cos(k|x− y|), (5.6)

for some K ∈ C([0, 2π]).In the following let ∆, ∆2 be the first and second forward-difference operators, i.e. for a

sequence akk∈N∆ak := ak+1 − ak

and ∆2 := ∆ ∆. Moreover, let

Dn(x) :=n∑

k=−neikx = 1 + 2

n∑k=1

cos(kx), x ∈ R

be the Dirichlet kernel and

Fn(x) :=n∑k=0

Dk(x), x ∈ R,

be the unnormalized Fejer kernel.

Lemma 5.2.1. Let akk∈N be such that ∆2(k2ak) ≤ 0, ∀k ∈ N and

limk→∞

k3|∆2ak|+ k2|∆ak|+ k|ak| = 0. (5.7)

Then

K(x) =a0

2+∞∑k=1

ak cos(kx)

exists locally uniformly in (0, 2π), is convex on [0, 2π] and decreasing on [0, π].

Proof. We first note that since

∆(k2ak) = k2∆ak + (2k + 1)ak+1

and∆2(k2ak) = k2∆2ak + 2(2k + 1)∆ak+1 + 2ak+2

assumption (5.7) is equivalent to

limk→∞

|k∆2(k2ak)|+ |∆(k2ak)|+ k|ak| = 0. (5.8)

We now follow ideas from [Kra11]. Using the Abel transformation we observe

Sn(x) =a0

2+

n∑k=1

ak cos(kx) =1

2

n∑k=0

∆ak+1Dk(x) +1

2an+1Dn(x).

3Actually, the Gaussian structure plays no role in this section. Our entire analysis applies whenever thecovariance function has the form (5.6).

114


By the assumptions and (5.8) we have∑∞

k=1 |∆ak| <∞. Since supn∈NDn(x) is bounded locallyuniformly on (0, 2π) and an → 0 we observe that

K(x) :=a0

2+∞∑k=1

ak cos(kx) =1

2

∞∑k=0

∆akDk(x)

exists locally uniformly and is continuous in (0, 2π).The Cesaro means of the sequence Sn(x) are given by

σn(x) =a0

2+

n∑k=1

(1− k

n+ 1

)ak cos(kx).

By Fejer’s Theorem [Zyg59, Theorem III.3.4] and continuity of K, σn → K locally uniformlyin (0, 2π). Hence, σ′′n → K ′′ in the space of distributions on (0, 2π). Clearly,

σ′′n(x) = −n∑k=0

(1− k

n+ 1

)k2ak cos(kx).

Let βk :=(

1− kn+1

)k2ak. Using summation by parts twice we obtain

2σ′′n(x) =

n∑k=0

∆βkDk(x)

= ∆βnFn(x)−n−1∑k=0

∆2βkFk(x)

= −n−1∑k=0

∆2(k2ak)Fk(x)−n−1∑k=0

(k∆2(k2ak)

n+ 1−

2∆((k + 1)2ak+1

)n+ 1

)Fk(x) +

n2

n+ 1anFn(x).

By (5.8) and 0 ≤ Fn(x) ≤ Cx2

+ C(2π−x)2

it follows

lim infn→∞

infx∈[ε,2π−ε]

σ′′n(x) ≥ 0,

for every ε > 0. For any non-negative test-function ϕ ∈ C∞c (0, 2π) Fatou’s Lemma implies

K ′′(ϕ) = limn→∞

σ′′n(ϕ) ≥∫ 2π

0lim infn→∞

σ′′n(x)ϕ(x)dx ≥ 0,

i.e. K ′′ is a non-negative distribution on (0, 2π). Thus, K is convex on [0, 2π]. Assume nowthat K is not decreasing on [0, π], i.e. there are x < y ∈ [0, π] such that K(x) < K(y). SinceK is given as a cosine series, we have K(x) = K(x′) and K(y) = K(y′) for x′ = 2π − x andy′ = 2π − y. Choose t ∈ (0, 1) such that tx+ (1− t)x′ = y. Then

K(tx+ (1− t)x′) = K(y) > K(x) = tK(x) + (1− t)K(x′)

which is in contradiction to the convexity of K.

So far we need to require concavity of the Fourier coefficients k2ak in order to control the2D-ρ variation of the corresponding covariance function RK . Smoothing RK should preservethe control of its 2D-ρ variation, while it does not have to preserve concavity of the Fouriercoefficients. The following Proposition shows that finiteness of the 2D-ρ variation is indeedpreserved under convolution with a measure of finite total variation and that it can be boundedby the total variation norm of this measure (cf. also [FV10b, Proposition 5.64]).

115


In the following let M(S1) be the space of signed, real Borel-measures on S1 with finitetotal variation ‖ · ‖TV . We say that a sequence µn ∈M(S1) weakly converges to µ ∈M(S1) if

µn(ϕ) :=

∫ 2π

0ϕ(x)dµn(x)→ µ(ϕ) =

∫ 2π

0ϕ(x)dµ(x), for n→∞,

for all 2π-periodic Lipschitz functions ϕ ∈ Lip([0, 2π]). Define Mw(S1) to be M(S1) endowedwith the topology of weak convergence. For B ∈ L1(S1) we set µB := B dx ∈M(S1) to be theassociated measure with density B.

Proposition 5.2.2. Assume that the covariance of Ψ is of the form

R(x, y) =∑k∈Z

akbkeik(x−y) (5.9)

with ak = a−k, bk = b−k real-valued for all k ∈ N and there is a real-valued measure µ ∈M(S1)such that

bk =

∫ 2π

0e−ikz µ(dz).

Moreover, assume that RK(x, y) =∑

k∈Z akeik(x−y) is of finite ρ-variation on [0, 2π]2 and∑

k∈Z |ak| <∞. Then R is of finite ρ-variation with

Vρ(R; [s, t]× [u, v]) ≤ ‖µ‖TV sup0≤z≤2π

Vρ(RK ; [s− z, t− z]× [u, v])

≤ 2‖µ‖TV Vρ(RK ; [0, 2π]2)

for every [s, t]× [u, v] ⊆ [0, 2π]2. Both estimates also hold for controlled ρ-variation.

Proof. Since∑

k∈Z |ak| <∞ we observe

R(x, y) =∑k∈Z

akbkeik(x−y) = (RK(·, y) ∗ µ) (x) =

∫ 2π

0RK(x− z, y) dµ(z)

Thus, using Jensen’s inequality,∣∣∣∣R( x, yx′, y′

)∣∣∣∣ρ ≤ (∫ 2π

0

∣∣∣∣RK ( x− z, y − zx′, y′

)∣∣∣∣ d|µ|(z))ρ≤ ‖µ‖ρTV

∫ 2π

0

∣∣∣∣RK ( x− z, y − zx′, y′

)∣∣∣∣ρ d |µ|(z)‖µ‖TV.

We will first show the estimate for the controlled ρ-variation. Let Π ∈ P ([s, t]× [u, v]). Then∑[x,y]×[x′,y′]∈Π

∣∣∣∣R( x, yx′, y′

)∣∣∣∣ρ ≤ ‖µ‖ρTV ∫ 2π

0

∑[x,y]×[x′,y′]∈Π

∣∣∣∣RK ( x− z, y − zx′, y′

)∣∣∣∣ρ d |µ|(z)‖µ‖TV

≤ ‖µ‖ρTV∫ 2π

0|RK |ρρ−var;[s−z,t−z]×[u,v]d

|µ|(z)‖µ‖TV

≤ ‖µ‖ρTV sup0≤z≤2π

|RK |ρρ−var;[s−z,t−z]×[u,v].

Taking the supremum over all partitions yields the first inequality. For the second inequality,we use periodicity to see that

|RK |ρρ−var;[s−z,t−z]×[u,v] ≤ |RK |ρρ−var;[−2π,2π]×[0,2π]

≤ 2ρ−1(|RK |ρρ−var;[−2π,0]×[0,2π] + |RK |ρρ−var;[0,2π]2

)≤ 2ρ|RK |ρρ−var;[0,2π]2

.

116


The estimate for the usual ρ-variation follows exactly in the same way by considering onlygrid-like partitions of [s, t]× [u, v], i.e. partitions of the form

[ti, ti+1]× [t′j , t′j+1] | (ti) partition of [s, t] and (t′j) partition of [u, v]

.

Remark 5.2.3. In many cases, z 7→ Vρ(RK ; [s− z, t− z]× [s, t]) attains its maximum at z = 0.If µ = δ0, we have bk = 1 for every k and the estimate above is sharp.

In order to use Proposition 5.2.2 to control the ρ-variation of R(x, y), we need to control‖µ‖TV . Since bk = b−k we observe

∑k∈Z

bkeikx = 2

(b02

+∞∑k=1

bk cos(kx)

).

Recall

Lemma 5.2.4. Let bkk∈Z be a sequence satisfying bk = b−k and bk → b ∈ R for k →∞ andlet Sn(x) := 1

2π

∑nk=−n bke

ikx. Assume one of the following conditions

(i)∑∞

k=1 |bk − b| <∞.

(ii) there exists a non-increasing sequence Ak such that∑∞

k=0Ak <∞ and |∆bk| ≤ Ak for allk ≥ 0.

(iii) bk is quasi-convex, i.e.∞∑k=0

(k + 1)|∆2bk| <∞.

Then, B(x) = 12π

∑k∈Z(bk − b)eikx locally uniformly on (0, 2π) and the right hand side is

the Fourier series of B. Moreover,

µSn µB + bδ0 =: µ, weakly in M(S1)

and bk =∫ 2π

0 e−ikz µ(dz). Moreover, there is a numerical constant C > 0 such that

‖µ‖TV ≤ |b|+ C

∑∞

k=1 |bk − b|, in case (1)∑∞k=1Ak, in case (2)∑∞k=0(k + 1)|∆2bk|, in case (3).

(5.10)

Proof. Step 1: We first restrict to the case b = 0.It is enough to prove that B(x) = 1

2π

∑k∈Z bke

ikx exists locally uniformly and in L1([0, 2π])with L1-bound corresponding to (5.10).

(1): obvious.(2): Assume that there exists a decreasing sequence Ak such that

∑∞k=0Ak < ∞ and

|∆bk| ≤ Ak for all k ≥ 0. In this case the claim has been proven in [Tel73]. For completenesswe include the proof.

Arguing as in the proof of Lemma 5.2.1 we obtain that

2πB(x) =b02

+

∞∑k=1

bk cos(kx) =1

2

∞∑k=0

∆bkDk(x)

exists locally uniformly in (0, 2π).

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Since∑∞

k=0Ak <∞ we have kAk → 0. Using summation by parts twice we obtain

b02

+∞∑k=1

bk cos(kx) =1

2

∞∑k=0

Ak∆bkAk

Dk(x) =1

2

∞∑k=0

|∆Ak|

(k∑i=0

∆biAi

Di(x)

).

Hence, ∫ 2π

0

∣∣∣∣∣b02 +∞∑k=1

bk cos(kx)

∣∣∣∣∣ dx ≤ 1

2

∞∑k=0

∆Ak

∫ 2π

0

∣∣∣∣∣k∑i=0

∆biAi

Di(x)

∣∣∣∣∣ dx.In [Tel73, Lemma 1] it is shown that there is a constant C > 0 such that for each sequenceaii∈N with |ai| ≤ 1 ∫ 2π

0

∣∣∣∣∣k∑i=0

aiDi(x)

∣∣∣∣∣ dx ≤ C(k + 1).

Therefore, ∫ 2π

0

∣∣∣∣∣b02 +

∞∑k=1

bk cos(kx)

∣∣∣∣∣ dx ≤ C∞∑k=0

(∆Ak)(k + 1) = C

∞∑k=0

Ak <∞.

Consequently, b02 +∑∞

k=1 bk cos(kx) ∈ L1([0, 2π]) and bi are the Fourier coefficients correspondingto B.

(3): Let now∑∞

k=0(k + 1)|∆2bk| < ∞. A direct proof of this classical result can be foundin [Kol23]. Here, we will prove that the assumptions in (i) are implied. We choose Ak :=∑∞

i=k |∆2bi|. Then∞∑k=0

Ak =

∞∑k=0

∞∑i=k

|∆2bi| =∞∑k=0

(k + 1)|∆2bk| <∞

and the claim follows from (2).

Step 2: Let now b ∈ R.

We split up Sn as

2πSn(x) =

n∑k=−n

bkeikx =

n∑k=−n

(bk − b)eikx + bDn(x).

By the first step we know that

2πB(x) =∑k∈Z

(bk − b)eikx

exists locally uniformly in (0, 2π) and in L1([0, 2π]) with bound on the L1-norm given by (5.10).Therefore, the measures µBn = 1

2π

∑nk=−n(bk − b)eikx dx converge to µB in Mw(S1). It is

well-known that Dn(x)→ 2πδ0 in Mw(S1). Thus,

Sn(x)→ µB + bδ0 =: µ, in Mw(S1).

The bound on the total variation norm of µ then follows from step one.

Lemma 5.2.4 in combination with Proposition 5.2.2 allows to derive bounds on the ρ-variation of covariance functions of type (5.9) depending on µ only via its total variation norm.Since we will use this to prove uniform estimates, we will need the following uniform estimateson the L1-norm of Fourier series.

118


Lemma 5.2.5. Let b ∈ C1(0,∞) with b(r)→ 0 for r →∞ and bτk := b(τmk) for some τ,m > 0.If

(i) b is convex, non-increasing, then bτk satisfies the assumptions of Lemma 5.2.4, (1), Bτ (x) =bτ02 +

∑∞k=1 b

τk cos(kx) exists locally uniformly in (0, 2π) and

‖Bτ‖L1([0,2π]) ≤ Cb0,

for some C > 0.

(ii) b ∈ C2(0,∞) with r|b′′(r)| ∈ L1(R+), then bτk satisfies the assumptions of Lemma 5.2.4,(2) and

‖Bτ‖L1([0,2π]) ≤ C∫ ∞

0r|b′′(r)|dr,

for some C > 0 with Bτ as in (1).

Proof. (1): Since b, |b′| are non-increasing ∆bτk ≤ 0 and −∆bk is non-increasing. We setAk := −∆bk. Clearly,

∑∞k=0Ak = 2b0 and the claim follows from Lemma 5.2.4.

(2): Let bτ (r) := b(τmr) and observe

∆2bτk = bτk+2 − 2bτk+1 + bτk =

∫ k+2

k+1(bτ )′ (s)ds−

∫ k+2

k+1(bτ )′ (s− 1)ds =

∫ k+2

k+1

∫ s

s−1(bτ )′′ (r)drds.

Since (bτ )′′ (r)dr = τmb′′(τmr)d(τmr), we obtain

∞∑k=0

(k + 1)|∆2bτk| ≤∞∑k=0

∫ k+2

k+1

∫ s

s−1(r + 1)| (bτ )′′ (r)|drds

≤∞∑k=0

∫ ∞0

∫ k+2

k+11[s−1,s](r)(r + 1)| (bτ )′′ (r)|dsdr

=

∫ ∞0

∫ ∞1

1[r,r+1](s)(r + 1)| (bτ )′′ (r)|dsdr

≤∫ ∞

0(r + 1)(r ∧ 1)| (bτ )′′ (r)|dr

=

∫ ∞0

τm(τ−mr + 1)(τ−mr ∧ 1)|b′′(r)|dr

=

∫ ∞0

r(τ−mr ∧ 1)|b′′(r)|dr +

∫ ∞0

(r ∧ τm)|b′′(r)|dr

≤ 2

∫ ∞0

r|b′′(r)|dr.

By Lemma 5.2.4 this finishes the proof.

Proposition 5.2.2 and Lemma 5.2.4 motivate the following partial order between boundedsequences

Definition 5.2.6. (i) A sequence (bk) is negligible if (bk) satisfies one condition (1) or (2)from Lemma 5.2.4.

(ii) A family of sequences (bτk) is uniformly negligible if (bτk) satisfies condition (1) or (2) fromLemma 5.2.4 with uniformly bounded right hand side in (5.10).

(iii) For two bounded sequences (ak),(ck) we write (ck) (ak) if there is a negligible sequence(bk) such that ck = akbk for every k.

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The reason for this definition is that as soon as we can control the ρ-variation of RK(x, y) =∑k∈Z ake

ik(x−y) and∑

k∈Z |ak| < ∞ we also have a control on the ρ-variation of R(x, y) =∑k∈Z cke

ik(x−y) for each sequence (ck) (ak) by Proposition 5.2.2 and Lemma 5.2.4. As animmediate Corollary we obtain

Proposition 5.2.7. Assume that the covariance of Ψ is of the form

R(x, y) =∑k∈Z

ckeik(x−y)

with ck ak and ak satisfying ak = a−k, ∆2(k2ak) ≤ 0 for all k ∈ N,

limk→∞

k3|∆2ak|+ k2|∆ak|+ k|ak| = 0

and (ak) is non-increasing with ak = O(k−(1+ 1

ρ))

for some ρ ≥ 1. Then the series K(x) =∑k∈Z ak cos(kx) is the Fourier series of a 1

ρ -Holder function, the covariance R of the random

process x 7→ (Ψ1(x), ...,Ψd(x)) is of finite ρ-variation and there is a constant C > 0 such that

Vρ(R; [x, y]2)ρ ≤ C|y − x|,

holds for all [x, y]2 ⊆ [0, 2π]2. The constant C depends only on ρ, the Holder-norm of K andthe right hand side of (5.10). If, in addition, ρ ∈ [1, 2) then x 7→ (Ψ1(x), ...,Ψd(x)) lifts to a

geometric p-rough path Ψ ∈ C0, 1p−Holder

0 ([0, 2π], G[p](Rd)) almost surely for every p > 2ρ.

Proof. We first consider the case ck = ak for all k ∈ N and verify the assumptions in Corollary5.1.4. By assumption x 7→ (Ψ1(x), ...,Ψd(x)) is a centered, continuous Gaussian process withindependent components. By Lemma 5.2.1 we know that

x 7→ σ2(x) = E|Ψ1(x)−Ψ1(0)|2 = 2(K(0)−K(x))

is concave on [0, 2π] and non-decreasing on [0, π]. Moreover, in [Lor48, Satz 8] it is shown that

K is 1ρ -Holder if and only if ak = O

(k−(1+ 1

ρ))

and in particular σ2(x) ≤ 2|K| 1ρ−Hol|x|

1ρ . Hence,

Theorem 5.1.2 and Corollary 5.1.4 yield the claim.

In the case of general sequences (bk) Proposition 5.2.2 together with Lemma 5.2.4 imply theclaim.

5.3 Lifting Ornstein-Uhlenbeck processes in space

Recall that we aim to construct spatial geometric rough path lifts of Ornstein-Uhlenbeck pro-cesses corresponding to

dΨit = (−(−∆)α − 1)Ψi

t dt+ dW it (5.11)

with periodic boundary conditions on [0, 2π]. Note that (−∆)α admits an orthonormal basis ofeigenvectors in L2([0, 2π]) which up to multiplicities is given by the Fourier basis

ek(x) :=

sin(kx), k > 012 , k = 0

cos(kx), k < 0.

120

Lifting Ornstein-Uhlenbeck processes in space

As a natural generalization of the fractional Laplacian (−∆)α in (5.11) we now consider theoperator A on L2([0, 2π]) given by

D(A) =

x ∈ L2([0, 2π])

∣∣∣∣ ∑k∈Z

µ2k(x, ek)

2 <∞

Ax =

∑k∈Z

µk(x, ek),

(5.12)

where (µk) is a sequence satisfying µ−k = µk and µk ≥ 0 for all k ∈ Z. The case of the(fractional) Laplace operator A = (−∆)α is recovered with the choice µk = |k|2α.

We consider (possibly) colored Wiener noise Wt with covariance operator Q given by Qek :=σkek, where σk is a non-negative sequence of real numbers satisfying σk = σ−k. Thus, (infor-mally) we can write

Wt =∑k∈N

√σkβ

kt ek,

for a sequence βk of independent Rd-valued standard Brownian motions.We will apply Corollary 5.1.4 to construct geometric rough path lifts of the strictly stationary

solution todΨi

t = (−A− λ)Ψitdt+ dW i

t , i = 1, ..., d. (5.13)

with λ > 0. Note that ek diagonalizes A+ λ with eigenvalues

λk = µk + λ ≥ λ > 0.

Assume from now on ∑k∈Z

σkλk

<∞. (5.14)

Then there is a unique strictly stationary mild solution Ψ to (5.13). The decomposition

Ψ(t, x;ω) =∑k∈Z

Y kt (ω)ek(x) =

∑k∈Z

Zkt (ω)eikx (5.15)

leads to a decoupled, infinite system of d-dimensional Ornstein-Uhlenbeck processes:

dY kt = −λkY k

t dt+√σkdβ

kt ,

and Zkt := 12

(Y−|k|t − isgn(k)Y

|k|t

)with sgn(0) := 0, for all k ∈ Z. The Y k are independent

Rd-valued Gaussian processes with stationary increments and i.i.d. components. Since λk = λ−kand σk = σ−k we have L(Y k) = L(Y −k) for all k ∈ N. In the following we consider a singlecomponent of Ψ and Y k and suppress the corresponding index for simplicity of notation. Note

EZkt Z ls = EY kt Y

ls = e−λk|t−s|

σk2λk

δk,l (5.16)

and thus

RK(x, y) = EΨ(t, x)Ψ(t, y) =1

2

∑k∈Z

σk2λk

eik(x−y) = K(x− y)

with

K(x) =σ0

4λ0+

∞∑k=1

σk2λk

cos(kx) =∑k∈Z

σk4λk

eikx.

121


Proposition 5.3.1. (i) Assume that(σkλk

) (k−2α) for some α ∈ (1

2 , 1] and that(σkλk

)is

eventually non-increasing. Then, for every t ≥ 0, the spatial process x 7→ Ψt(x) is astationary, centered Gaussian process which admits a continuous modification (which wedenote by the same symbol). Moreover, the covariance RK is of finite ρ-variation for allρ ≥ 1

2α−1 and

Vρ(RK ; [x, y]2) ≤ C|x− y|1ρ

holds for all [x, y]2 ⊆ [0, 2π]2 and for each t ∈ [0, T ]. If α > 34 , the process x 7→ Ψt(x) lifts

to a geometric p-rough path Ψ(t) ∈ C0, 1p−Holder

0 ([0, 2π], G[p](Rd)) almost surely for everyp > 2

2α−1 .

(ii) Assume that (σk) is bounded, (λk) is eventually non-decreasing4 and that

|∆λk|λk

= O

(1

k

). (5.17)

Then there is a continuous (even Holder continuous) modification of the map

Ψ : [0, T ]→ C0, 1p−Holder

0 ([0, 2π], G[p](Rd))t 7→ Ψ(t).

(5.18)

Proof. (1): It is clear that the spatial processes are stationary, centered and Gaussian. Con-

cerning continuity, note that since negligible sequences are bounded,(σkλk

) (k−2α) implies

that there is a constant C such that∣∣∣σkλk ∣∣∣ ≤ C

∣∣ 1k

∣∣2α holds for all k. Since(σkλk

)is eventually

non-increasing, there is a non-increasing sequence (ak) and a sequence (bk) for which bk = 0 forall k ≥ N for some N ∈ N, and σk

λk= ak + bk. It follows that there is a constant C1 such that

|ak| ≤ C1

∣∣ 1k

∣∣2α and [Lor48, Satz 8] implies that the corresponding cosine series defines 2α − 1Holder continuous function. Hence also K is the Fourier series of a 2α − 1 Holder continuousfunction. Therefore,

E |Ψt(x)−Ψt(y)|2 = 2|K(0)−K(x− y)| . |x− y|2α−1

which implies that there is a continuous modification. Hence we can apply Proposition 5.2.7which yields the claim.

(2): We will derive the existence of a continuous modification by application of Kolmogorov’scontinuity theorem. Therefore, we need an estimate on a q-th moment of the distance in thed 1p−Hol metric of the rough paths Ψ(t), Ψ(s) at different times 0 ≤ s < t ≤ T . Such an estimate

can be obtained by applying [FV10b, Theorem 15.37]: Let 0 ≤ s ≤ t ≤ T , τ := |t − s| andX(x) := (Ψ1(t, x), ...,Ψd(t, x)), Y (x) := (Ψ1(s, x), ...,Ψd(s, x)). We will first prove that thecovariance of (X,Y ) has finite Holder dominated ρ′-variation for all ρ′ > 1

2α−1 , uniformly in τ .Note that

Rτ (x, y) = EX1(x)Y 1(y) = EΨ1(t, x)Ψ1(s, y)

=∑k∈Z

σk4λk

e−λkτ cos(k(x− y))

=∑k∈Z

(σk|k|2α

4λk

)(|k|−2εαe−λkτ

)|k|−2(1−ε)α cos(k(x− y))

4These conditions may be relaxed in various ways. However, we decided to formulate it like that for the sakeof simplicity.

122


for every ε > 0. We will apply Proposition 5.2.2 multiple times. We therefore need to proofthat the first two sequences are (uniformly) negligible. For the first sequence this follows fromour assumptions. We will show that the ∆-criterion holds for the sequence (|k|−2εαe−τλk). Bythe mean value theorem,

|∆e−τλk | ≤ |∆λk|λk ∧ λk+1

supξ>0|ξe−ξ| . |k|−1.

Therefore,

|∆(|k|−2εαe−τλk)| ≤ |∆|k|−2εα|+ |k|−2εα|∆e−τλk | . |k|−1−2εα.

Since this is summable for α, ε > 0 we can apply Lemma 5.2.4 which shows that the secondsequence is uniformly negligible. This implies that Rτ has finite Holder dominated ρ′ variationfor ρ′ = 1

2(1−ε)α−1 , uniformly in τ > 0, for every ε > 0. Choose ε > 0 small enough such that

p > 2ρ′ holds. Then [FV10a, Theorem 37] implies∣∣∣d 1p−Hol(Ψ(t),Ψ(s))

∣∣∣Lq≤ C√q|RΨ(t)−Ψ(s)|θ∞,

for some θ = θ(p, ρ′) > 0 and all q ∈ [1,∞). In order to estimate the right hand side we note

|RΨ1(t)−Ψ1(s)(x, y)| = |EΨ1(t, x)Ψ1(t, y)−Ψ1(t, x)Ψ1(s, y)

−Ψ1(t, y)Ψ1(s, x) + Ψ1(s, x)Ψ1(s, y)|

=

∣∣∣∣∣∑k∈Z

σk2λk

(1− e−λkτ

)eik(x−y)

∣∣∣∣∣≤∑k∈N

σkλk

∣∣∣1− e−λkτ ∣∣∣≤ C

∑k≤N

σk|t− s|+ CN1−2α′∑k>N

k2α′−1σkλk

≤ C(N |t− s|+N1−2α′)

for α′ ∈ (1/2, α). We then choose N ∼ |t− s|−1

2α′ to obtain

|RΨ1(t)−Ψ1(s)(x, y)| ≤ C|t− s|1−1

2α′ ,

and thus ∣∣∣d 1p−Hol(Ψ(t),Ψ(s))

∣∣∣Lq≤ C√q|t− s|θ(1−

12α′ ),

for some θ > 0 and all p > 22α−1 , q ∈ [1,∞). Kolmogorov’s continuity Theorem proves that for

small γ > 0 there is a modification of t 7→ Ψ(t) (again denoted by Ψ(t)) such that(E‖Ψ‖qγ−Hol;[0,T ]

) 1q ≤ C <∞,

for some constant C = C(γ, T, ρ, θ, q).

Example 5.3.2 We consider the stochastic fractional heat equation with (possibly) colorednoise on the 1-dimensional torus, i.e.

dΨit = (−AαΨi

t − λ) dt+ dA−γ2W i

t , i = 1, ..., d, (5.19)

123


where Aα = (−∆)α, α ∈ (0, 1], γ ≥ 0, λ > 0 and Wt is a cylindrical Wiener process. Hence,

λk = |k|2α + λ and σk = |k|−2γ . We claim that(σkλk

) (k−(2γ+2α)). To see this, we need to

show that(|k|2α

λ+|k|2α

)is negligible. Set g(x) = 1

λx+1 . By the mean value theorem,∣∣∣∣∆( |k|2α

λ+ |k|2α

)∣∣∣∣ ≤ |g′|∞ ∣∣|k + 1|−2α − |k|−2α∣∣ . |k|−2α−1.

Since this is summable the claim is shown for α > 0. Therefore, the assumptions of Proposition5.3.1 (1) are satisfied if 2γ + 2α > 3

2 . Clearly,

|∆λk|λk

=|∆|k|2α||k|2α + λ

∼ |k|2α−1|k|−2α ∼ |k|−1,

and (5.17) holds. Thus, there is a geometric p-rough path (1p -Holder continuous) Ψt, lifting the

strictly stationary solution Ψt to (5.19) in space for every p > 22γ+2α−1 and t ≥ 0. Moreover,

there is a continuous modification of the map

Ψ : [0, T ]→ C0, 1p−Holder

0 ([0, 2π], G[p](Rd)).

3.1 Stability and approximationsIn this section we will consider hyper-viscosity approximations and Galerkin approximations

to Ψ and prove the strong convergence of the corresponding rough paths lifts.The hyper-viscosity approximation Ψε = (Ψε,1, . . . ,Ψε,d) is the solution to

dΨε,it = (−A− εAβ − λ)Ψε,i

t dt+ dW it , i = 1, ..., d, (5.20)

for some (large) β ≥ 1 and ε > 0. The Ψε are called (hyper-)viscosity approximations of Ψ. Asbefore, ek diagonalizes (A+ εAβ + λ) with eigenvalues

λεk = µk + εµβk + λ > 0.

The covariance Rε of every component of Ψε(t) is given by Rε(x, y) = Kε(|x− y|) where

Kε(x) =∑k∈Z

σk cos(kx)

4λεk.

The Galerkin approximation ΨNt of Ψt is defined to be the projection of Ψ onto the (2N+1)-

dimensional subspace spanned by ek|k|≤N . This process solves the SPDE

dΨN,it = (−PNA− λ)ΨN,i

t dt+ dPNWit , i = 1, ..., d (5.21)

where PNA has the eigenvalues µk1|k|≤N and PNWt has the covariance operator QN given by

QNek = 1|k|≤Nσkek. The covariance RN of ΨN,i(t) is of the form RN (x, y) = KN (|x−y|) where

KN (x) =∑|k|≤N

σk cos(kx)

4λk.

One easily checks that we can lift the spatial sample paths of Ψεt and ΨN

t to Gaussian roughpaths and find continuous modifications of t 7→ Ψε

t resp. t 7→ ΨNt . Moreover, we can prove the

following strong convergence result:

124


Proposition 5.3.3. Under the same assumptions as in Proposition 5.3.1, for every ρ > 12α−1

there are constants C1, C2 such that

supε>0

Vρ(R(Ψt,Ψεt ); [x, y]2) ≤ C1|x− y|

1ρ and sup

N∈NVρ(R(Ψt,ΨNt ); [x, y]2) ≤ C2|x− y|

1ρ (5.22)

holds for every square [x, y]2 ⊆ [0, 2π]2 and t ∈ [0, T ]. Moreover, for every p > 22α−1 , q > 0 and

t ∈ [0, T ], one has∣∣∣d 1p−Hol(Ψt,Ψ

εt )∣∣∣Lq(P)

→ 0 and∣∣∣d 1

p−Hol(Ψt,Ψ

Nt )∣∣∣Lq(P)

→ 0, (5.23)

for ε→ 0 resp. N →∞.

Proof. Hyper-viscosity approximation: To prove (5.22), we have to show that

supε>0

Vρ(Rε; [x, y]2) . |x− y|1/ρ and sup

ε>0Vρ(f

ε; [x, y]2) . |x− y|1/ρ (5.24)

holds for every [x, y]2 where f ε(x, y) = E(

Ψit(x)Ψε,i

t (y))

. In order to get a control on the ρ-

variation of Rε we apply Proposition 5.2.2 and Lemma 5.2.4 multiple times. To do so, we needto decompose the coefficients σk

4λεkas a product of a well-controlled sequence (ak) and possibly

multiple negligible coefficients. For simplicity of notation we write λk := λ0k. We note

σk4λεk

= |k|−2(1−δ)α(|k|2α σk

4λk

)(|k|−2δαλk

λεk

).

We need to show that the second and the third sequence are uniformly negligible. For the

second sequence, this follows by our assumptions. Note that λkλεk

=

(1 + ε

µβkλk

)−1

and by the

mean value theorem,

∣∣∣∣∆(λkλεk)∣∣∣∣ ≤ sup

ξ>0

ξ

(1 + ξ)2

∣∣∣∣∆(µβkλk

)∣∣∣∣µβkλk∧ µβk+1

λk+1

∼ µ1−βk

∣∣∣∣∣∆(µβkλk

)∣∣∣∣∣ .Furthermore,

|∆λ−1k | ≤

|∆λk|λ2k

≤ |∆µk|µ2k

and thus ∣∣∣∣∣∆(µβkλk

)∣∣∣∣∣ ≤ µβk |∆λ−1k |+ λ−1

k+1|∆µβk | . µβ−2

k |∆µk|+ µ−1k |∆µ

βk |,

which shows that ∣∣∣∣∆(λkλεk)∣∣∣∣ . |∆µk|µk

+|∆µβk |µβk

∼ |∆λk|λk

= O(k−1),

uniformly in ε. Since∣∣∣λkλεk ∣∣∣ ≤ 1,∣∣∣∣∆(|k|−2δαλk

λεk

)∣∣∣∣ ≤ |∆|k|−2δα|+ |k|−2δα

∣∣∣∣∆(λkλεk)∣∣∣∣ = O(k−1−2δα),

125


which shows that for every δ > 0,(|k|−2δα λk

λεk

)is uniformly negligible. For any ρ > 1

2α−1 , we

can choose δ > 0 small enough such that ρ = 12(1−δ)α−1 holds. This shows the left hand side of

(5.24). For Ψit(x) =

∑k∈Z Y

kt ek(x) and Ψε,i

t (x) =∑

k∈Z Yk,εt ek(x) we have

EY kt Y

l,εt = E

∫ 0

−∞eλks√σk dβ

ks

∫ 0

−∞eλ

εl s√σl dβ

ls

= σk

∫ ∞0

e−(λk+λεk)s ds δk,l =σk

λk + λεkδk,l.

Hence

f ε (x, y) =∑k∈Z

ek(x)ek(y)EY kt Y

k,εt

=∑k∈Z

ek(x)ek(y)σk

λk + λεk

=σ0

2(λ0 + λε0)+

∞∑k=1

σkλk + λεk

cos(k(x− y)).

We decompose the coefficients as follows:

σ

λk + λεk= |k|−2(1−δ)α

(|k|2ασk

λk

)(|k|−2δα λk

2λk + εµβk

)

for δ > 0. Noting that λk2λk+εµβk

=

(2 + ε

µβkλk

)−1

, we can proceed as above to see that also the

right hand side of (5.24) holds and thus (5.22) is shown in the hyper-viscosity case. It remainsto prove (5.23). Using [FV10a, Theorem 37] and the Cauchy Schwarz inequality, it is enoughto show that

supx∈[0,2π]

E|Ψt(x)−Ψεt (x)|2 → 0 (5.25)

for ε→ 0. We have

E|Ψ(x, t)−Ψε(x, t)|2 = E|Ψ(x, t)|2 + E|Ψε(x, t)|2 − 2EΨ(x, t)Ψε(x, t)

≤∑k

∣∣∣∣ σk4λk+

σk4λεk− σkλk + λεk

∣∣∣∣ .Now ∣∣∣∣ σk4λk

+σk4λεk− σkλk + λεk

∣∣∣∣ ≤ σk4λk

+σk

4λk + 4εµβk+

σk

2λk + εµβk≤ σkλk

and since this is summable, we can use dominated convergence to see that indeed (5.25) holdsfor ε→ 0.

Galerkin approximation: Again we have to prove that

supN∈N

Vρ(RN ; [x, y]2) . |x− y|1/ρ and sup

N∈NVρ(f

N ; [x, y]2) . |x− y|1/ρ (5.26)

holds where fN (x, y) = EΨN,it (x)Ψi

t(y). Note that

KN (x) =∑k∈Z|k|−2(1−δ)α

(|k|2α σk

4µk

)(|k|−2δα

1|k|≤N ) cos(kx)

126


for all δ > 0. The second sequence is negligible by assumption. Using Proposition 5.2.2, itsuffices to show that the Fourier series

BN =∑k∈Z|k|−2δα

1|k|≤Nek =∑|k|≤N

|k|−2δαek

is uniformly bounded in L1([0, 2π]). Since ∆k−2δα = O(k−2δα−1) and limk→∞ log(k)k−2δα = 0we can apply the Sidon-Telyakovskii Theorem (cf. [Tel73, Theorem 4]) to obtain BN → B forN → ∞ in L1([0, 2π]). This proves the left hand side of (5.26). If ΨN,i

t (x) =∑

k∈Z Ykt ek(x)

with Y kt = 1|k|≤NY

kt , one has

fN (x, y) =∑k∈Z

ek(x)ek(y)EY kt Y

kt =

∑|k|≤N

ek(x)ek(y)E|Y kt |2 =

∑|k|≤N

cos(k(x− y))σk4µk

= KN (x− y).

With the first part, this implies (5.26) and thus (5.22) in the case of the Galerkin approximations.Furthermore,

E|Ψ(x, t)−ΨN (x, t)|2 ≤∑k∈Z

σk4µk

+∑|k|≤N

σk4µk− 2

∑|k|≤N

σk4µk

=∑|k|>N

σk4µk→ 0

for N →∞ due to summability.

Remark 5.3.4. One can check, using Lemma 5.2.5, that in the special case µk = |k|ζ we have

supε>0

Vρ(R(Ψt,Ψεt ); [x, y]2) ≤ C1|x− y|

1ρ

even for ρ = 12α−1 for all squares [x, y]2 ⊆ [0, 2π]2.

Remark 5.3.5. The bounds obtained in (5.22) can also be used for other purposes. For instance,they become crucial when proving uniform (exponential) integrability of certain related stochasticintegrals, a question raised in [Hai11]. See [FR13] for further details.

Remark 5.3.6. Our calculations are well suited to determine also the rate of convergencein (5.23) for the inhomogeneous rough paths metric ρ 1

p−Hol(·, ·) (cf. [FV10b] for the formal

definition of these metrics). As an example, we consider the Galerkin approximations. Since∑|k|>N

σk2µk

.∑|k|>N

1

|k|2α∼∫ ∞N

x−2α dx,

we have E|Ψ(x, t)−ΨN (x, t)|2 → 0 for N →∞ with rate 2α− 1. Using the results of [FR] (seealso [RX12]), for every ε > 0 there is a p = pρ,ε > 2ρ such that∣∣∣% 1

p−Hol(Ψt,Ψ

Nt )∣∣∣Lq(P)

. q12b 1pc

supx∈[0,2π]

(E|Ψ(x, t)−ΨN (x, t)|2

)1− ρ2−ε.

Hence for every ε > 0, we can choose ρ close enough to 12α−1 to see that∣∣∣% 1

p−Hol(Ψt,Ψ

Nt )∣∣∣Lq(P)

→ 0,

for N →∞ with rate 2α− 32 − ε. Using a Borel-Cantelli argument, we also obtain almost sure

convergence with the same rate. Note that in general, one has to choose p large in order toobtain the optimal convergence rate (cf. [FR] for the optimal choice of p).

127


3.2 The continuous case

Consider the stationary solution of

dΨt = −((−∆)α + λ)Ψt dt+ dWt, on R,

for some α ∈ (0, 1], λ > 0. The stationary solution can be written down explicitly (cf. [Wal86]),namely

Ψt(x) =

∫ t

−∞

∫RKt−s(x, y)W (ds, dy),

whereK is the fractional heat kernel operator associated to−((−∆)α+λ) with Fourier transformgiven by

Kt(ξ) = e−t|ξ|2α−λt.

After some calculations, one sees that the covariance R of the spatial process x 7→ Ψt(x) forevery time point t is given by R(x, y) = K(x− y) where

K(x) =

∫ ∞−∞

f(ξ)e−ixξ dξ

and

f(ξ) =1

2|ξ|2α + 2λ.

In order to deduce the existence of a rough path lift of x 7→ Ψt(x) on compact intervals ofR by means of Theorem 5.1.2 we have to prove convexity of K, which will follow from Lemma5.3.7 below if α > 1

2 . It is easy to see that σ2(x) = E(Ψt(x)−Ψt(0))2 . |x|2α−1. Hence, we canapply Corollary 5.1.4 to see that Ψt can be lifted, for every fixed time point t, to a process Ψt

with sample paths in C0,β−Holder0 (I,G[1/β](Rd)), every β < α− 1/2, provided α > 3/4, where I

can be an arbitrary compact interval in R containing 0.

It remains to give a continuous version of Lemma 5.2.1 proving convexity of K, i.e. a criterionfor the convexity of Fourier transforms. For a function f ∈ L1(R) we define

f(x) =

∫ ∞−∞

f(ξ)e−ixξ dξ.

Then the following Lemma holds:

Lemma 5.3.7. Assume that f , f ∈ L1(R), f(ξ) = f(−ξ) for all ξ, f is twice differentiablealmost everywhere,

limξ→∞

ξ3|f ′′(ξ)|+ ξ2|f ′(ξ)|+ ξ|f(ξ)| = 0

and that there is an x0 ∈ (0,∞] such that

lim supR→∞

∫ R

0

∂2

∂ξ2(f(ξ) ξ2)Fξ(x) dξ ≤ 0,

for all x ∈ (0, x0) where Fξ(x) = 1−cos(ξx)x2

denotes the Fejer kernel. Then f is a convex functionon [0, x0).

128


Proof. Since the proof is very similar to Lemma 5.2.1 we just sketch it briefly. By Fejer’sTheorem for Fourier transforms (cf. [Kor89, Theorem 49.3]),

limR→∞

1

2π

∫ R

−R

(1− |ξ|

R

)g(ξ)eixξ dξ = g(x),

for all x provided g ∈ C ∩ L1. Setting g = f , we obtain from Fourier inversion

limR→∞

∫ R

−R

(1− |ξ|

R

)f(ξ)eixξ dξ = f(x).

We then proceed as in the proof of Lemma 5.2.1.

Note that given f ∈ L1 it does not follow in general that also f ∈ L1. However, Bernstein’sTheorem states that the Fourier transform of functions f in the Sobolev space Hs are containedin L1 for all s > 1

2 (cf. [Hor83, Corollary 7.9.4]).

129


130

6

From rough path estimates tomultilevel Monte Carlo

We consider implementable schemes for large classes of stochastic differential equations (SDEs)

dYt = V (Yt) dXt (ω)

driven by multidimensional Gaussian signals, say X = Xt (ω) ∈ Rd. The interpretation ofthese equations is in Lyons’ rough path sense [LQ02, LCL07, FV10b]. This requires smooth-ness/boundedness conditions on the vector fields V = (V1, . . . , Vd); for simplicity, the readermay assume bounded vector fields with bounded derivatives of all order (but we will be morespecific later). This also requires a “natural” lift of X (·, ω) to a (random) rough path X· (ω),a situation fairly well understood, cf [FV10b, Ch. 15] and the references therein. For instance,fractional Brownian motion [CQ02] is covered for Hurst parameter H > 1/4. It may help thereader to recall that, in the case when X = B, a multidimensional Brownian motion, all thisamounts to enhance B with Levy’s stochastic area or, equivalently, with all iterated stochasticintegrals of B against itself, say Bs,t =

∫ ts Bs,· ⊗ dB. The (rough-)pathwise solution concept

then agrees with the usual notion of an SDE solution (in Ito- or Stratonovich sense, dependingon which integration was used in defining B). As is well-known this provides a robust exten-sion of the usual Ito framework of stochastic differential equations with an exploding numberof new applications (including non-linear SPDE theory, robustness of the filtering problem,non-Markovian Hormander theory).

In a sense, the rough path interpretation of a differential equation is most closely related tostrong, pathwise error estimates of Euler- resp. Milstein-approximation to stochastic differentialequations. For instance, Davie’s definition [Dav07] of a (rough)pathwise SDE solution is

Yt − Ys ≡ Ys,t = Vi (Ys)Bis,t + V k

i (Ys) ∂kVj (Ys)Bi,js,t + o (|t− s|) as t ↓ s. (6.1)

In fact, this becomes an entirely deterministic definition, only assuming

∃α ∈ (1/3, 1/2) : |Bs,t| ≤ C |t− s|α , |Bs,t| ≤ C |t− s|2α ,

something which is known to hold true almost surely (i.e. for C = C (ω) < ∞ a.s.), andsomething which is not at all restricted to Brownian motion. As the reader may suspect thisapproach leads to almost-sure convergence (with rates) of schemes which are based on theiteration of the approximation seen in the right-hand-side of (6.1). The practical trouble is thatLevy’s area, the antisymmetric part of B, is notoriously difficult to simulate; leave alone thesimulation of Levy’s area for other Gaussian processes. It has been understood for a while,at least in the Brownian setting, that the truncated (or: simplified) Milstein scheme, in whichLevy’s area is omitted, i.e. replace Bs,t by Sym (Bs,t) in (6.1), still offers benefits: For instance,Talay [Tal86] replaces Levy area by suitable Bernoulli r.v. such as to obtain weak order 1 (see

Rough paths estimates and MLMC

also Kloeden–Platen [KP92] and the references therein).1 In the multilevel context, [GS12] usethis truncated Milstein scheme together with a sophisticated antithetic (variance reduction)method. Finally, in the rough path context this scheme was used in [DNT12]: the convergenceof the scheme can be traced down to an underlying Wong-Zakai type approximation for thedriving random rough path – a (probabilistic!) result which is known to hold in great generalityfor stochastic processes, starting with [CQ02] in the context of fractional Brownian motion, see[FV10b, Ch. 15] and the references therein.

A rather difficult problem is to go from almost-sure convergence (with rates) to L1 (orever: Lr any r < ∞) convergence. Indeed, as pointed out in [DNT12, Remark 1.2]: ”Notethat the almost sure estimate [for the simplified Milstein scheme] cannot be turned into anL1-estimate [...]. This is a consequence of the use of the rough path method, which exhibitsnon-integrable (random) constants.” The resolution of this problem forms the first contributionof this chapter. It is based on some recent progress [CLL, FR13], initially developed to provesmoothness of laws for (non-Markovian) SDEs driven by Gaussian signals under a Hormandercondition, [CF10, HP].

Having established Lr-convergence (any r <∞, with rates) for implementable “simplified”Milstein schemes we move to our second contribution: a multilevel algorithm, in the sense ofGiles [Gil08b], for stochastic differential equations driven by large classes of Gaussian signals.The key remark here is that there is not much downside to replace the weak error estimate (“rateα”) which forms part of Giles’ abstract condition [Gil08b, Theorem 3.1], by the correspondingstrong estimate. Indeed, a strong, L2 error estimate (“rate β/2”) is the key assumption in Giles’complexity theorem, and this is precisely what we have established in the first part. Some otherextension of the Giles theorem are necessary; indeed it is crucial to allow for α < 1/2 (ruledout explicitly in [Gil08b]) whenever we deal with driving signals with sample path regularity“worse” then Brownian motion. Luckily this can be done without too much trouble.

More precisely, we consider the following scheme for approximating Y , see [DNT12, FV10b].Given a time-grid 0 = t0 < t1 < · · · < tn = T , the corresponding increments Xtk,tk+1

, k =0, . . . , n− 1, of the driving noise and a (suitably big) integer N ∈ N, define Y 0 ≡ Y0 and

Y tk+1= Y tk +

N∑l=1

1

l!Vi1 · · ·VilI

(Y tk

)Xi1tk,tk+1

· · ·Xiltk,tk+1

, (6.2)

where I(y) = y is the identity function, V = (V1, . . . , Vd) for vector fields V1, . . . , Vd, whichhave also the interpretation as linear first order operators, acting on functionals by Vig(y) =∇g(y) · Vi(y). Moreover, the Einstein summation convention is in force. For a more detaileddescription of the algorithm we refer to Section 6.2. Combining this discretization scheme withmulti-level Monte Carlo simulation, we obtain the following main result:

Theorem 6.0.1. Assume the Gaussian driving signal has independent components, with Holderdominated covariance of finite ρ-variation (in the precise sense of [FV10b, Ch. 15], [FV11]; ex-amples include multi-dimensional Brownian motion with ρ = 1 and fractional Brownian motionwith Hurst parameter H ∈ (1/4, 1/2], with ρ = 1/ (2H)). Let f : C([0, T ],Rm) → Rn be a Lip-schitz continuous functional. Then the Monte Carlo evaluation of a path-dependent functionalof the form E(f(Y·)), Y being the solution of an RDE driven by this Gaussian signal, to withina MSE of ε2, can be achieved with computational complexity

O(ε−θ)∀θ > 2ρ

2− ρ.

In the case of SDEs driven by Brownian motion (ρ = 1) our computational complexity

is arbitrarily ”close” to known result O(ε−2 (log ε)2

)[Gil08a, Gil08b], recently sharpened to

O(ε−2)

[GS12] with the aid of a suitable antithetic multilevel correction estimator.

1A well-known counter-example by Clark and Cameron [CC80]) shows that it is impossible to get strong order1 if only using Brownian increments.

132

Rough path estimates revisited

A direct Monte Carlo implementation of the scheme (6.2) would require a complexity ofO(ε−(2+1/α) in order to attain an MSE of no more than ε2. Here, α is the weak rate of conver-gence of the scheme. This rate clearly depends on the regularity of the functional, but under onlyweak regularity conditions, the rate is equal to the strong rate (under stronger regularity condi-tions, the weak rate can be significantly bigger). In this case, the number of time-steps neededto guarantee a weak discretization error E[f(YT )]−E[f(Y T )] = O(ε) is O(ε−1/α), whereas thenumber of samples needed to guarantee a statistical error of order O(ε) is O(ε−2), which givesthe claimed complexity bound of O(ε−(2+1/α)). Consequently, the use of multilevel Monte Carloin this highly degenerate case gives an amazing boost in the efficiency, as the complexity canbe reduced by a factor ε.

6.1 Rough path estimates revisited

1.1 Preliminaries

Definition 6.1.1. Let ω be a control. For α > 0 and [s, t] ⊂ [0, T ] we set

τ0 (α) = s

τi+1 (α) = inf u : ω (τi, u) ≥ α, τi (α) < u ≤ t ∧ t

and defineNα,[s,t] (ω) = sup n ∈ N∪0 : τn (α) < t .

When ω arises from the (homogeneous) p-variation norm ‖ · ‖p−var of a (p -rough) path, x, i.e.ωx = ‖x‖pp-var;[·,·], we shall also write Nα,[s,t] (x) := Nα,[s,t] (ωx).

Recall that if ω1 and ω2 are controls, also ω1 + ω2 is a control.

Lemma 6.1.2. Let ω1 and ω2 be two controls. Then

Nα,[s,t](ω1 + ω2) ≤ 2Nα,[s,t](ω

1) + 2Nα,[s,t](ω2) + 2

for every s < t and α > 0.

Proof. If ω is any control, set

ωα (s, t) := sup(ti)=D⊂[s,t]ω(ti,ti+1)≤α

∑ti

ω(ti, ti+1).

If ω := ω1 + ω2, ω(ti, ti+1) ≤ α implies ωi(ti, ti+1) ≤ α for i = 1, 2 and therefore ωα (s, t) ≤ω1α (s, t)+ω2

α (s, t). From Proposition 4.6 in [CLL] we know that ωiα (s, t) ≤ α(2Nα,[s,t]

(ωi)

+ 1)

for i = 1, 2. (Strictly speaking, Proposition 4.6 is formulated for a particular control ω, namelythe control induced by the p-variation of a rough path. However, the proof only uses generalproperties of control functions and the conclusion remains valid.) We conclude

αNα,[s,t] (ω) =

Nα,[s,t](ω)−1∑i=0

ω(τi (α) , τi+1 (α))

≤ ωα(s, t)

≤ ω1α (s, t) + ω2

α (s, t)

≤ α(2Nα,[s,t]

(ω1)

+ 2Nα,[s,t]

(ω2)

+ 2).

133


Lemma 6.1.3. Let ω1 and ω2 be two controls and assume that ω2(s, t) ≤ K. Then

Nα,[s,t](ω1 + ω2) ≤ Nα−K,[s,t](ω

1)

for every α > K.

Proof. Set ω := ω1 + ω2 and

τ0 (α) = s

τi+1 (α) = inf u : ω (τi, u) ≥ α, τi (α) < u ≤ t ∧ t.

Similarly, we define (τi)i∈N = (τi(α − K))i∈N for ω1. It suffices to show that τi ≥ τi fori = 0, . . . , Nα,[s,t](ω). We do this by induction. For i = 0, this is clear. If τi ≥ τi for somei ≤ Nα,[s,t](ω)− 1, superadditivity of control functions gives

α = ω(τi, τi+1) ≤ ω1(τi, τi+1) +K

which implies τi+1 ≤ τi+1.

Let x1,x2 be p-rough paths and ω a control. Let V i = (V i1 , . . . , V

id ), i = 1, 2 be two families

of vector fields, γ > p, ν a bound on |V 1|Lipγ and |V 2|Lipγ and y1, y2 the solutions of the RDEs

dyit = V i(yit) dxit; yis ∈ Re

for s ≤ t and i = 1, 2.

Lemma 6.1.4. Let s < t ∈ [0, T ] and assume that ‖xi‖p−ω;[s,t] ≤ 1 for i = 1, 2. Then there isa constant C = C(γ, p) such that

ν|y1 − y2|∞;[s,t] ≤[ν|y1

s − y2s |+

∣∣V 1 − V 2∣∣Lipγ−1 + νρp−ω;[s,t](x

1,x2)]

× (Nα,[s,t](ω) + 1) expCνpα(Nα,[s,t](ω) + 1)

for every α > 0.

Proof. Set y = y1 − y2 and

κ =

∣∣V 1 − V 2∣∣Lipγ−1

ν+ ρp−ω;[s,t](x

1,x2).

From [FV10b, Theorem 10.26] we can deduce that there is a constant C = C(γ, p) such that

|yu,v| ≤ Cνω(u, v)1/p [|yu|+ κ] exp Cνpω(u, v)

for every u < v ∈ [s, t]. From |yu,v| ≥ |ys,v| − |ys,u| we obtain

|ys,v| ≤ Cνω(u, v)1/p [|yu|+ κ] exp Cνpω(u, v)+ |ys,u|≤ [|ys|+ |ys,u|+ κ] exp Cνpω(u, v)

for s ≤ u < v ≤ t. Now let s = τ0 < τ1 < . . . < τM < τM+1 = v ≤ t for M ≥ 0. By induction,one sees that

|ys,v| ≤ (M + 1)(|ys|+ κ) exp

Cνp

M∑i=0

ω(τi, τi+1)

≤ CM+1 [|ys|+ κ] exp

Cνp

M∑i=0

ω(τi, τi+1)

.

134

Rough path estimates revisited

It follows that for every v ∈ [s, t],

|ys,v| ≤ [|ys|+ κ] (Nα,[s,t](ω) + 1) expCνpα(Nα,[s,t](ω) + 1)

,

therefore|yv| ≤ [|ys|+ κ] (Nα,[s,t](ω) + 1) exp

Cνpα(Nα,[s,t](ω) + 1)

+ |ys|

and finally

|y|∞;[s,t] ≤ [|ys|+ κ] (Nα,[s,t](ω) + 1) expCνpα(Nα,[s,t](ω) + 1)

.

Next, we recover the well-known local Lipschitz continuity property of the Ito Lyons map.In comparison with [FV10b, Theorem 10.26], the estimate is sharpened in a way that we canuse the integrability results of [CLL]..

Corollary 6.1.5. Consider the RDEs

dyit = V i(yit) dxit; yi0 ∈ Re

for i = 1, 2 on [0, T ] where V 1 and V 2 are two families of vector fields, γ > p and ν is a boundon |V 1|Lipγ and |V 2|Lipγ . Then for every α > 0 there is a constant C = C(γ, p, ν, α) such that∣∣y1 − y2

∣∣∞;[0,T ]

≤ C[|y1

0 − y20|+

∣∣V 1 − V 2∣∣Lipγ−1 + ρp−var;[0,T ](x

1,x2)]

× expC(Nα,[0,T ](x

1) +Nα,[0,T ](x2) + 1

)holds.

Proof. Let ω be a control such that ‖xi‖p−ω;[0,T ] ≤ 1 for i = 1, 2 (the precise choice of ω will bemade later). From 6.1.4 we know that there is a constant C = C(γ, p, ν, α) such that∣∣y1 − y2

∣∣∞;[0,T ]

≤[|y1

0 − y20|+

∣∣V 1 − V 2∣∣Lipγ−1 + ρp−ω;[s,t](x

1,x2)]

× expC(Nα,[s,t](ω) + 1)

.

Now we set ω = ωx1,x2 where

ωx1,x2(s, t) = ‖x1‖pp−var;[s,t] + ‖x2‖pp−var;[s,t] +

bpc∑k=1

(ρ

(k)p−var;[s,t](x

1,x2))p/k

(ρ

(k)p−var;[0,T ](x

1,x2))p/k .

It is easy to check that

‖x1‖p−ωx1,x2 ;[0,T ] ≤ 1,

‖x2‖p−ωx1,x2 ;[0,T ] ≤ 1 and

ρp−ωx1,x2 ;[0,T ](x1,x2) ≤ ρp−var;[0,T ](x

1,x2).

Finally, if α > bpc we can use Lemma 6.1.3 and Lemma 6.1.2 to see that

Nα,[0,T ](ωx1,x2) + 1 ≤ Nα−bpc,[0,T ](ωx1 + ωx2) + 1

≤ 3(Nα−bpc,[0,T ](x

1) +Nα−bpc,[0,T ](x2) + 1

).

Substituting α 7→ α+ bpc gives the claimed estimate.

135


Remark 6.1.6. Comparing the result above with [FV10b, Theorem 10.26], one sees that weobtain a slightly weaker result; namely, the distance between y1 and y2 is measured here inuniform topology instead of p-variation topology. However, with little more effort, one can showthat the same estimate holds for ρp-var; [0,T ]

(y1, y2

)instead of

∣∣y1 − y2∣∣∞;[0,T ]

.

1.2 Deterministic convergence of Euler approximations based on entire rough path

We are now interested in convergence rates for Euler schemes. Recall the notation from[FV10b]: If V = (V1, . . . , Vd) is a collection of sufficiently smooth vector fields on Re, g ∈TN

(Rd)

and y ∈ Re, we define an increment of the step-N Euler scheme by

E(V ) (y, g) :=N∑k=1

Vi1 . . . VikI (y) gk,i1,...,ik

where gk,i1,...,ik = πk (g)i1,...,ik ∈ R, I is the identity on Re and every Vj is identified with

the first-order differential operator V(k)j (y) ∂

∂yk(throughout, we use the Einstein summation

convention). Furthermore, we set

Egy := y + E(V ) (y, g) .

Given D = 0 = t0 < . . . < tn = T and a path x ∈ Cp−var([0, T ] ;Gbpc

(Rd))

we define the(step-N) Euler approximation to the RDE solution y of

dy = V (y) dx (6.3)

with starting point y0 ∈ Re at time tk ∈ D by

yEuler;Dtk

:= Etk←t0y0 := ESN (x)tk−1,tk · · · ESN (x)t0,t1y0.

Proposition 6.1.7. Let x ∈ Cp−var([0, T ] ;Gbpc

(Rd))

and set ω (s, t) = ‖x‖pp−var;[s,t]. Assume

that V ∈ Lipθ for some θ > p and let ν ≥ |V |Lipγ . Choose N ∈ N such that bpc ≤ N ≤θ. Fix a dissection D = 0 = t0 < . . . < tn = T of [0, T ] and let yEuler;DT denote the step-

N Euler approximation of y. Then for every ζ ∈[Np ,

N+1p

)and α > 0 there is a constant

C = C (p, θ, ζ,N, ν, α) such that∣∣∣yT − yEuler;DT

∣∣∣ ≤ C expC(Nα,[0,T ](x) + 1

) n∑k=1

ω (tk−1, tk)ζ .

In particular, if x is a Holder rough path and |tk+1 − tk| ≤ |D| for all k we obtain∣∣∣yT − yEuler;DT

∣∣∣ ≤ CT ‖x‖ζp1/p-Hol;[0,T ] expC(Nα,[0,T ](x) + 1

)|D|ζ−1 (6.4)

Proof. We basically repeat the proof of [FV10b, Theorem 10.30]. Recall the notation π(V ) (s, ys; x)for the (unique) solution of (6.3) with starting point ys at time s. Set

zk = π(V )

(tk,E

tk←t0y0; x).

Then z0t = yt, z

ktk

= Etk←t0y0 for every k = 1, . . . , n and znT = yEuler;DT , hence

∣∣∣yT − yEuler;DT

∣∣∣ ≤ n∑k=1

∣∣∣zkT − zk−1T

∣∣∣ .136

Probabilistic convergence results for RDEs

One can easily see that

zk−1T = π(V )

(tk−1, z

k−1tk−1

; x)

= π(V )

(tk, z

k−1tk

; x)

for all k = 1, . . . , n. Applying Corollary 6.1.5 (in particular the Lipschitzness in the startingpoint) we obtain for any α > 0∣∣∣zkT − zk−1

T

∣∣∣ ≤ c1

∣∣∣zktk − zk−1tk

∣∣∣ expc1

(Nα,[0,T ](x) + 1

).

Moreover (cf. [FV10b, Theorem 10.30]),∣∣∣zktk − zk−1tk

∣∣∣ ≤ ∣∣∣π(V ) (tk−1, ·,x)tk−1,tk− E(V )

(·, SN (x)tk−1,tk

)∣∣∣∞.

Let δ ∈ [0, 1) such that ζ = N+δp . Since (N + δ) − 1 < N ≤ γ we have V ∈ Lip(N+δ)−1. Thus

we can apply [FV10b, Corollary 10.15] to see that

∣∣∣π(V ) (tk−1, ·,x)tk−1,tk− E(V )

(·, SN (x)tk−1,tk

)∣∣∣∞≤ c2

(|V |Lip(N+δ)−1 ‖x‖p−var;[tk−1,tk]

)N+δ

≤ c2 |V |pζLipγ ω (tk−1, tk)ζ


6.2 Probabilistic convergence results for RDEs

2.1 Lr-rates for step-N Euler approximation (based on entire rough path)

We now give convergence rates for the step-N Euler scheme in Lp. Although the schemedescribed here is not easy to implement (simulation of higher order iterated integrals!) it willserve as a stepping stone to establish the rates for the (easy-to-implement) simplified Eulerscheme discussed later in this section.

For simplicity, we formulate it only in the Holder case.

Theorem 6.2.1. Let X : [0, T ]→ Rd be a continuous, centered Gaussian process with indepen-

dent components. Assume further that Vρ

(RX ; [s, t]2

)≤ K |t− s|1/ρ holds for all s < t, some

ρ ∈ [1, 2) and a constant K. Let H denote the Cameron-Martin space associated to X. Assumethat

ι : H → Cq−var

and let M ≥ |ι|op. Choose p > 2ρ and assume that V ∈ Lipθ for some θ > p and let ν ≥|V |Lipθ . Set D = 0 < ε < 2ε < . . . < (bT/εc − 1)ε < T and let Y Euler;D

T denote the step-NEuler approximation of Y , the (pathwise) solution of

dY = V (Y ) dX ; Y0 ∈ Re

where N is chosen such that bpc ≤ N ≤ θ. Then for every r ≥ 1, r′ > r and ζ ∈[Np ,

N+1p

)there is a constant C = C(ρ, p, q, θ, ν,K,M, r, r′, N, ζ) such that∣∣∣YT − Y Euler;D

T

∣∣∣Lr≤ CT

∣∣∣‖X‖ζp1/p-Hol;[0,T ]

∣∣∣Lr′

εζ−1

holds for all ε > 0.

137


Remark 6.2.2. By choosing p ∈ (2ρ, p) one has N+1p < N+1

p and applying the Theorem with pinstead of p shows that ∣∣∣YT − Y Euler;D

T

∣∣∣Lr

. εN+1p−1

holds for every p > 2ρ if ε→ 0.

Proof. Similar to the proof of the forthcoming Theorem 6.2.4: We use the pathwise estimate(6.4) and take the Lr norm on both sides. The Holder inequality shows that∣∣∣YT − Y Euler;D

T

∣∣∣Lr≤ c1T

∣∣∣‖X‖ζp1/p-Hol;[0,T ]

∣∣∣Lr′

∣∣expC(Nα,[0,T ](X) + 1

)∣∣Lr′′|D|ζ−1

holds for some (possibly large) r′′ > r. Applying the results of [CLL] (see also [FR13]) showsthat

∣∣expC(Nα,[0,T ](X) + 1

)∣∣Lr′′

< ∞ holds for all α > 0. However, if we want a bounddepending only on K and M , we have to (and can!) choose α large enough (using the result of[FR13]) to obtain ∣∣exp

C(Nα,[0,T ](X) + 1

)∣∣Lr′′≤ c2 <∞.

2.2 Lr-rates for Wong-Zakai approximations

We aim to formulate a version of the Wong-Zakai Theorem which contains convergence ratesin Lr, any r ≥ 1 for a class of suitable approximations Xε of X. By this, we mean that

(i) (Xε, X) : [0, T ] → Rd+d is jointly Gaussian,(Xε;i, Xi

)and

(Xε;j , Xj

)are independent

for i 6= j and

supε∈(0,1]

Vρ

(R(Xε,X); [0, T ]2

)=: K <∞

for some ρ ∈ [1, 2).

(ii) Uniform convergence of the second moments:

supt∈[0,T ]

E[|Xε

t −Xt|2]

=: δ (ε)1/ρ → 0 for ε→ 0.

Example 6.2.3 A typical example of such approximations are the piecewise linear ap-proximations of X (ω) at the time points 0 < ε < 2ε < . . . < (bT/εc − 1)ε < T (see [FV10b,

Chapter 15.5]). In the case Vρ

(RX ; [s, t]2

). |t− s|1/ρ (i.e. if we deal with Holder rough paths),

one can show that δ (ε) . ε.

Theorem 6.2.4. Let X : [0, T ] → Rd be a centered Gaussian process with continuous samplepaths and independent components and Xε a suitable approximation as above for some ρ ∈[1.2). Let Hε denote the Cameron-Martin space of the joint process (X,Xε). Assume thatcomplementary Young regularity holds for the sample paths of (X,Xε) and the paths in theCameron Martin space Hε, i.e. there are p, q ≥ 1 such that p > 2ρ, 1

p + 1q > 1 and that there is

a continuous embedding

ιε : Hε → Cq−var

and furthermore

supε∈(0,1]

|ιε|op =: M <∞.

138

Probabilistic convergence results for RDEs

Let X and Xεdenote the lift of X resp. Xε to a process with p-rough sample paths for somep > 2ρ. Let V = (V1, . . . , Vd) be a collection of vector fields in Re with |V |Lipθ ≤ ν < ∞ where

θ ≥ 2ρρ−1 . Let Y, Y ε : [0, T ]→ Re denote the pathwise solutions to the equations

dYt = V (Yt) dXt; Y0 ∈ Re

dY εt = V (Y ε

t ) dXεt; Y ε

0 = Y0 ∈ Re.

Then, for any η < 1ρ −

12 and r ≥ 1 there is a constant C = C(ρ, p, q, θ, ν,K,M, η, r) such that∣∣∣|Y ε − Y |∞;[0,T ]

∣∣∣Lr≤ Cδ (ε)η

holds for all ε > 0.

Remark 6.2.5. The assumptions on the Cameron-Martin paths are always fulfilled for ρ ∈[1, 3/2) using the Cameron-Martin embedding from [FV10b]. In this case, M ≤

√K. In the case

ρ ∈ [3/2, 2), they are still fulfilled provided complementary Young-regularity holds for the samplepaths of X and the paths in its Cameron-Martin space H and if the operators Λε : ω 7→ ωε areuniformly bounded as operators Cq−var → Cq−var. This is the case, for instance, when dealingwith piecewise-linear or mollifier approximations.

Proof. Set X0 = X and let Hε denote the Cameron-Martin space associated to Xε. By assump-tion, we know that

|h|q−var ≤M |h|Hε

holds for every h ∈ Hε and ε ≥ 0. Lemma 5 together with Corollary 2 and Remark 1 in [FR13]show that there is a α = α(p, ρ,K) > 0 and a positive constant c1 = c1 (p, q, ρ,M) such that wehave the uniform tail estimate

P (Nα,[0,T ](Xε) > u) ≤ exp

−c1α

2/pu2/q

for all u > 0 and ε ≥ 0. Choose γ such that η = 12ρ

(1− ρ

γ

). By assumption on η we have

1ρ + 1

γ > 1 and we can choose p ∈ (2γ, θ). Set Xε = Sbpc (Xε) and X = Sbpc (X). Lipschitznessof the map Sbpc and [FR13, Lemma 2] show that also

P (Nα,[0,T ](Xε) > u) ≤ exp

−c1α

2/pu2/q

(6.5)

hold for all u > 0 and ε ≥ 0 for a possibly smaller α > 0. Now we use Corollary 6.1.5 and theCauchy-Schwarz inequality to see that∣∣∣|Y ε − Y |∞;[0,T ]

∣∣∣Lr≤ c2

∣∣∣ρp−var;[0,T ](Xε, X)

∣∣∣L2r

∣∣∣expc2

(Nα,[0,T ](X

ε) +Nα,[0,T ](X) + 1)∣∣∣

L2r

for a constant c2. The uniform tail estimates (6.5) show that

supε≥0

∣∣∣expc2

(Nα,[0,T ](X

ε) +Nα,[0,T ](X) + 1)∣∣∣

L2r≤ c3 <∞.

Using [FR, Theorem 6] gives∣∣∣ρp−var;[0,T ](Xε, X)

∣∣∣L2r≤ c4 sup

t∈[0,T ]|Xε

t −Xt|1− ρ

γ

L2 = δ (ε)η

for a constant c4 which finishes the proof.

139


2.3 Lr-rates for the simplified Euler schemes

For N ≥ 2, step-N Euler schemes contain iterated integrals whose distributions are not easyto simulate when dealing with Gaussian processes. In contrast, the simplified step-N Eulerschemes avoid this difficulty by substituting the iterated integrals by a product of increments.In the context of fractional Brownian motion, it was introduced in [DNT12]. We make thefollowing definition: If V = (V1, . . . , Vd) is sufficiently smooth, x is a p-rough path, y ∈ Re andN ≥ bpc, we set

Esimple(V )

(y, SN (x)s,t

):=

N∑k=1

1

k!Vi1 . . . VikI (y)xi1s,t · · ·x

iks,t

for s < t and

ESN (x)s,tsimple y := y + Esimple

(V )

(y, SN (x)s,t

).

Given D = 0 = t0 < . . . < tn = T and a path x ∈ Cp−var([0, T ] ;Gbpc

(Rd))

we define thesimplified (step-N) Euler approximation to the RDE solution y of

dy = V (y) dx

with starting point y0 ∈ Re at time tk ∈ D by

ysimple Euler;Dtk

:= Etk←t0simpley0 := ESN (x)tk−1,tk

simple · · · ESN (x)t0,t1simple y0

and at time t ∈ (tk, tk+1) by

ysimple Euler;Dt :=

(t− tk

tk+1 − tk

)(ysimple Euler;Dtk+1

− ysimple Euler;Dtk

)+ ysimple Euler;D

tk.

Corollary 6.2.6. Let X : [0, T ] → Rd be as in Theorem 6.2.1. Assume that |V |Lipθ ≤ ν < ∞for some θ ∈ (0,∞] chosen such that θ ≥ 2ρ

ρ−1 . Choose N ∈ N such that b2ρc ≤ N ≤ θ andD = 0 < ε < 2ε < . . . < (bT/εc − 1)ε < T for ε > 0. Then for any δ > 0 and r ≥ 1,∣∣∣∣∣∣Y − Y simple Euler;D

∣∣∣∞

∣∣∣Lr

. ε12ρ−δ

+ ε1ρ− 1

2−δ

+ εN+12ρ−1−δ

for all ε > 0.

Remark 6.2.7. In the proof we will see that the rate 12ρ − δ comes from the (almost) 1

2ρ -

Holder-regulartiy of the sample paths of Y , the rate 1ρ −

12 − δ is the rate for the Wong-Zakai

approximation and N+12ρ − 1− δ comes from the step-N Euler approximation. Since we always

assume ρ ≥ 1, the Wong-Zakai error always dominates the first error. In particular, for ρ = 1we can choose N = 2 to obtain a rate arbitrary close to 1

2 . For ρ > 1, the choice N = 3 gives arate of almost 1

ρ −12 . In both cases the rate does not increase for larger choices of N .

Proof. Let Xε denote the Gaussian process whose sample paths are piecewise linear approxi-mated at the time points given by D and let Y ε : [0, T ] → Re denote the pathwise solution tothe equation

dY ε = V (Y ε) dXε; Y ε0 = Y0 ∈ Re.

Then for any tk, tk+1 ∈ D we have

Xε;k;i1,...,iktk,tk+1

=1

k!Xi1tk,tk+1

· · ·Xiktk,tk+1

,

140

Giles’ complexity theorem revisited

hence Y simple Euler;Dt = Y ε; Euler;D

t for any t ∈ D and thus∣∣∣Yt − Y simple Euler;Dt

∣∣∣ ≤ |Y − Y ε|∞ + maxtk∈D

∣∣∣Y εtk− Y ε; Euler;D

tk

∣∣∣if t ∈ D. For t /∈ D, choose tk ∈ D such that tk < t < tk+1. Set a = t−tk

tk+1−tk and b =tk+1−ttk+1−tk .

Then a+ b = 1 and by the triangle in equality,∣∣∣Yt − Y simple Euler;Dt

∣∣∣ ≤ a∣∣Yt − Ytk+1

∣∣+ b |Yt − Ytk |+ a∣∣∣Ytk+1

− Y simple Euler;Dtk+1

∣∣∣+ b

∣∣∣Ytk − Y simple Euler;Dtk

∣∣∣. ε1/p ‖Y ‖1/p-Hol;[0,T ] + max

tk∈D

∣∣∣Ytk − Y simple Euler;Dtk

∣∣∣. ε1/p

(‖X‖1/p-Hol;[0,T ] ∨ ‖X‖

p1/p-Hol;[0,T ]

)+ |Y − Y ε|∞

+ maxtk∈D

∣∣∣Y εtk− Y ε; Euler;D

tk

∣∣∣for any p > 2ρ. Since the right hand side does not depend on t, we can pass to the sup-normon the left hand side. We now take the Lr-norm on both sides and check that the conditions ofTheorem 6.2.4 and 6.2.1 are fulfilled and that the constants can be chosen independently fromε. Since we are dealing with piecewise linear approximations, we have

supε>0

Vρ

(R(Xε,X); [0, T ]2

)<∞ and sup

t∈[0,T ]E[|Xε

t −Xt|2]. ε1/ρ

(cf. [FV10b, Chapter 15]). Furthermore, for every ω ∈ Ω one has |ωε|p−var ≤ 31−1/p |ω|p−var and

|ωε|1/p−Hol ≤ 31−1/p |ω|1/p−Hol for every p ≥ 1 and ε > 0 (follows, for instance, from [FV10b,Theorem 5.23]). This shows that we can apply Theorem 6.2.4 to see that for any δ > 0,

||Y − Y ε|∞|Lr . ε1ρ− 1

2−δ

holds for all ε > 0. Furthermore, can choose p′ > 2ρ such that N+1p′ −1 = N+1

2ρ −1−δ and then ap-

ply Theorem 6.2.1. Since∣∣d1/p′-Hol(X

ε,X)∣∣Lr→ 0 for ε→ 0, clearly supε>0

∣∣∣‖Xε‖1/p′-Hol;[0,T ]

∣∣∣Lr<

∞ and the constants on the right hand side do indeed not depend on ε. Choosing p such that1p = 1

2ρ − δ gives the claim.

6.3 Giles’ complexity theorem revisited

We adapt the main theorem of [Gil08b] to our later needs. Below one should think

P = f (Y·)

for a Lipschitz function f and Y the solution to the Gaussian RDE dY = V (Y ) dX. Let Pldenote some (modified) Milstein approximation a la [DNT12], for instance (6.2), based on ameshsize hl = T/(M0M

l). Recall the basic idea

E [P ] ≈ E[PL

]for L large

= E[P0

]+

L∑l=1

E[Pl − Pl−1

]

141


set P−1 ≡ 0 and define the (unbiased) estimator Yl of E[Pl − Pl−1

], say

Yl =1

Nl

Nl∑i=1

(P

(i)l − P

(i)l−1

)(6.6)

based on i = 1, . . . , Nl independent samples. Note that P(i)l − P

(i)l−1 comes from approximations

with different mesh but the same realization of the driving noise. In fact, we have the abstracttheorem, an extension of [Gil08b] to the case α < 1/2.

Theorem 6.3.1. Let 0 < α < 1/2 and 0 < β ≤ 2α. Following Giles, we assume that there areconstants c1, c′2, c2 and c3 such that

(i) E[Pl − P

]≤ c1h

αl ,

(ii) E[Y0

]= E

[P0

]and E

[Yl

]= E

[Pl − Pl−1

], l > 0,

(iii) V[Y0

]≤ c′2N

−10 and V

[Yl

]≤ c2N

−1l hβl for l ∈ N,2

(iv) the complexity Cl of computing Yl is bounded by C0 ≤ c3N0h−10 for l = 0 and Cl ≤

c3Nl(h−1l + h−1

l−1) for l ≥ 1.3

Then for every ε > 0, there are choices L and Nl, 0 ≤ l ≤ L, to be given below in (6.10)and (6.11), respectively, and constants c4 and c5 given in (6.12) together with (6.13) such thatthe multilevel estimator Y =

∑Ll=0 Yl satisfies the mean square error bound

MSE ≡ E[(Y − E[P ]

)2]< ε2,

with complexity bound

C ≤ const ε−1+2α−β

α + o(ε−

1+2α−βα

),

where const = c4 for β < 2α and const = c4 + c5 for β = 2α.

Proof. We first ignore the basic requirement of L and Nl being integer values to obtain (almost)optimal real-valued choices for L and Nl. Then we are going to verify the above given boundsfor the MSE and the complexity using the smallest integers dominating our real-valued choices.In this proof, we abuse notation by setting T = T/M0, noting that both complexity and MSEonly depend on T and M0 by T/M0.

2We distinguish between c′2 and c2, since the former controls the variance V[Y0

], which is often already

proportional to the variance of f(Y·), whereas the latter controls the variance of the difference Yl, which is oftenmuch smaller in size.

3Note that the complexity at the 0-level is proportional to the number of timesteps h−10 , whereas at higher

levels, we need to apply the numerical scheme twice, once for the finer and once for the coarser grid.

142


The mean-square-error satisfies

MSE = E

[(Y − E[P ]

)2]

= E

[(Y − E

[Y])2

+(E[Y]− E[P ]

)2]

= V[Y]

+(E[PL

]− E[P ]

)2

≤L∑l=0

V[Yl

]+ c2

1h2αL

≤ c′2N−10 + c2T

βL∑l=1

N−1l M−lβ + c2

1h2αL .

Now we need to minimize the total computational work

C ≤ c3N0h−10 + c3

L∑l=1

Nl

(h−1l + h−1

l−1

)= c3T

−1

[N0 +

M + 1

M

L∑l=1

NlMl

]

under the constraint MSE ≤ ε2. We first assume L to be given and minimize over N0, . . . , NL,and then we try to find an optimal L. We consider the Lagrange function

f(N0, . . . , NL, λ) ≡ c3T−1

[N0 +

M + 1

M

L∑l=1

NlMl

]+

+ λ

(c′2N

−10 + c2T

βL∑l=1

N−1l M−lβ + c2

1h2αL − ε2

).

Taking derivatives with respect to Nl, 0 ≤ l ≤ L, we arrive at

∂f

∂N0= c3T

−1 − λc′2N−20 = 0,

∂f

∂Nl= c3T

−1M lM + 1

M− λc2T

βM−lβN−2l = 0,

implying that

N0 =√λ

√c′2c3T , (6.7a)

Nl =√λ

√c2

c3T (1+β)/2

√M

M + 1M−l(1+β)/2, 1 ≤ l ≤ L, (6.7b)

which we insert into the bound for the MSE to obtain

√λ =

[√c′2c3T−β +

√c2c3

√M + 1

MM (1−β)/2M

L(1−β)/2 − 1

M (1−β)/2−1

]T−(1−β)/2

ε2 − c21T

2αM−2αL. (6.8)

143


By construction, we see that for any such choice of N0, . . . , NL, the MSE is, indeed, boundedby ε2. For fixed L, the total complexity is now given by

C(L) ≡ c3T−1

√λ√c′2c3T +

M + 1

M

√c2c3

L∑l=1

M l√λ

√c2

c3T (1+β)/2

√M

M + 1M−l(1+β)/2

=√λT−(1−β)/2

[√c′2c3T−β +

√c2c3

√M + 1

MM (1−β)/2M

L(1−β)/2 − 1

M (1−β)/2 − 1

](6.9)

=

[√c′2c3T−β +

√c2c3

√M + 1

MM (1−β)/2M

L(1−β)/2 − 1

M (1−β)/2 − 1

]2T−(1−β)

ε2 − c21T

2αM−2αL.

In general, the optimal (but real-valued) choice of L would now be the arg-min of the abovefunction, which we could not determine explicitly in an arbitrary regime. We parametrize theoptimal choice of L by d1 in

L =

⌈log(d1c1T

αε−1)

α log(M)

⌉. (6.10)

There are three different approaches to the choice of L: Giles chooses d1 = 1/2, which is probablymotivated by the considerations in Remark 6.3.3 below. If all the constants involved have alreadybeen estimated, then one could choose L by numerical minimization of the complexity, or onecould provide an asymptotic optimizer L (for ε → 0). The latter approach has been carriedout for the special case β = 1 in Theorem 6.3.4 below, and in this case the optimal L is indeed(almost) of the form (6.10) with d1 weakly depending on ε.

Moreover, we choose with κ = 1−β2α

N0 =

⌈ √c′2T

β

ε2(1− d−21 )

(√c′2T

−β +√c2

√M + 1

MM (1−β)/2d

κ1cκ2T

(1−β)/2ε−κ − 1

M (1−β)/2 − 1

)⌉, (6.11a)

Nl =

√c2

√MM+1T

β

ε2(1− d−21 )

(√c′2T

−β +√c2

√M + 1

MM (1−β)/2d

κ1cκ2T

(1−β)/2ε−κ − 1

M (1−β)/2 − 1

)M−l(1+β)/2

,(6.11b)

1 ≤ l ≤ L.

By construction, the MSE will be bounded by ε2 using the choices (6.10) and (6.11). Asx ≤ dxe ≤ x+ 1 and using the inequalities

ML ≤ d1/α1 c

1/α1 TMε−1/α,

ML(1−β)/2 ≤ dκ1cκ1T (1−β)/2M (1−β)/2ε−κ,

together with the shorthand-notations

e1 =√c′2T

−β −√c2

√M + 1

M

M (1−β)/2

M (1−β)/2 − 1,

e2 = dκ1cκ+1/22

√M + 1

M

M (1−β)/2

M (1−β)/2 − 1T (1−β)/2

144


motivated from our choice (6.11), we arrive after a tedious calculation at

C ≤ c3T−1

[N0 +

M + 1

M

L∑l=1

NlMl

]

≤ c3T−1

[1 +

√c′2T

β(1− d−21 )−1(e1 + e2ε

−κ)ε−2+

+√c2T

β(1− d−21 )−1

√M + 1

M

M (1−β)/2

M (1−β)/2 − 1

(dκ1c

κ1T

(1−β)/2M (1−β)/2ε−κ − 1)

(e1 + e2ε−κ)ε−2+

+M + 1

M − 1

(d

1/α1 c

1/α1 TMε−1/α − 1

)].

Arranging the terms according to powers of ε and recalling κ = 1−β2α , we get

C ≤ c4ε−2(1+κ) + c5ε

−1/α + c6ε−(2+κ) + c7ε

−2 + c8, (6.12)

where

c4 = cκ1c1+κ2 c3

d2κ1

1− d−21

M + 1

M

M3(1−β)/2(M (1−β)/2 − 1

)2 , (6.13a)

c5 = c1/α1 c3d

1/α1

M(M + 1)

M − 1, (6.13b)

c6 = (cκ1 + cκ2)√c2c3

d2κ1

1− d−21

T−(1−β)/2

√M + 1

M

M (1−β)/2

M (1−β)/2 − 1(6.13c)

×

(√c′2T

−β −√c2

√M + 1

M

M (1−β)/2

M (1−β)/2 − 1

), (6.13d)

c7 =c3T

−(1−β)

1− d−21

e21, (6.13e)

c8 = −2c3T

−1

M − 1. (6.13f)

We remark that, under the condition that β ≤ 2α, we have 2(1 + κ) ≥ 1/α with equalityiff β = 2α. Consequently, ε−2(1+κ) is the dominating term in the complexity-expansion. Wefurther note that the second term in the expression can be either ε−1/α or ε−(2+κ).

The leading order coefficients c4 and c5 are positive, whereas the sign of c6 is not clear.In particular, if we do not distinguish between the variance of Y0 (controlled by c′2) and the

variances of the differences Yl, l = 1, . . . , L, controlled by c2, then c6 will be negative. c7 is againpositive (but often small) and c8 is negative. Clearly, we could simplify the complexity boundby omitting all terms with negative coefficients in (6.12). We further note that the leading orderterms of the complexity do not depend on T or M0.

We do not know the rate of weak convergence α of our simplified Milstein scheme, but forLipschitz functions f , we clearly have

|E [f(X·)− f(Y·)]| ≤ |f |LipE [|X − Y |∞] ,

so that the weak rate of convergence is at least as good as the strong rate of convergence, i.e.,α ≥ β/2 in the above notation. In fact, if we only impose minimal regularity conditions on f ,then it is highly unlikely that we can get anything better than α = β/2.

145


Corollary 6.3.2. Under the assumptions of Theorem 6.3.1, let us additionally assume thatα = β/2. Then the complexity of the above multi-level algorithm is bounded by

C ≤ c′4ε−1/α + o(ε−1/α

),

where

c′4 = c3d1/α1 (M + 1)

[c

(1−β)/β1 c

1/β2

d21 − 1

M3(1−β)/2

M(M (1−β)/2 − 1

)2 + c2/β1

M

M − 1

].

The optimal choice of d1 minimizing c′4 is obtained by

d1 =

√1− (1− β)f1

2f2+

√(1− β)2f2

1 + 4βf1f2

2f2,

with

f1 = c(1−β)/β1 c

1/β2 c3

M + 1

M

M3(1−β)/2(M (1−β)/2 − 1

)2 ,f2 = c

2/β1 c3

M(M + 1)

M − 1.

Proof. We use c′4 = c4 + c5 in order to obtain the formula for the constant. Then we consider c′4as a function of d1 and get the minimizer as the unique zero of the derivative in ]1,∞[, notingthat c′4 approaches ∞ on both boundaries of the domain.

Remark 6.3.3. In the now classical works of Giles on multilevel Monte Carlo, he usuallychooses d1 =

√2, see for instance [Gil08b].. This means that we reserve the same error tolerance

ε/2 both for the bias or discretization error and for the statistical or Monte Carlo error. In manysituations, this choice is not optimal. In fact, even in an ordinary Monte Carlo framework, oneshould not blindly follow this rule.

For instance, for an SDE driven by a Brownian motion, the Euler scheme usually (i.e.,under suitable regularity conditions) exhibits weak convergence with rate 1. Assuming the sameconstants for the weak error and the statistical error, a straightforward optimization will showthat it is optimal to choose the number of timesteps and the number of Monte Carlo samplessuch that the discretization error is ε/3 and the statistical error is 2ε/3.

In the above, the choice of d1 corresponds to the distribution of the total MSE ε2 betweenthe statistical and the discretization error according to

ε2 =ε2

d21︸︷︷︸

disc. error

+

(1− 1

d21

)ε2︸︷︷︸

stat. error

.

So, depending on the parameters, Corollary 6.3.2 shows that the canonical error distribution isnot optimal.

As the leading order coefficients c′4 depends only mildly on M , we do not try to find anoptimal choice of the parameter M .

The above analysis has also given us new insight into the classical multi-level Monte Carloalgorithm corresponding to the choice β = 1. Indeed, even in this case an equal distribution ofthe error tolerance ε among the bias and the statistical error is far from optimal. Indeed, wehave

146


Theorem 6.3.4. For β = 1 and α ≥ 1/2, the optimal choice of L is (apart from rounding up)given by

L(ε) =1

α logMlog

ε−1

c21T

2α

1 +

√c′2c2T

M

M + 1α logM

+ c21T

2α log ε−1

1/2

+O(log log ε−1

log ε−1),

which is of the form (6.10) with

d1 ≈

1 +

√c′2c2T

M

M + 1α logM + log ε−1

1/2

.

Proof. Let us investigate the behaviour of (6.9) for β ↑ 1. For β ↑ 1 we obtain

Cβ=1(L) =

[√c′2c3T−1 +

√c2c3

√M + 1

ML

]21

ε2 − c21T

2αM−2αL, (6.14)

and we want to minimize this object for L. Let us abbreviate to

D(L) :=C(L)

c′2c3T−1= (1 + aL)2 1

ε2 − bM−cL

with obvious definitions for a, b, c. Setting the derivative to zero yields

D′(L) = 2a (1 + aL)1

ε2 − bM−cL− (1 + aL)2 bcM

−cL logM

(ε2 − bM−cL)2 = 0

i.e.

2a(ε2 − bM−cL

)= (1 + aL) bcM−cL logM

2aε2M cL = 2ab+ bc logM + abcL logM

abbreviate again

ε2M cL = b+bc

2alogM +

bc

2L logM =: p+ qL

and write the latter asM cL

p+ qL=

1

ε2(6.15)

We now derive an asymptotic expansion for the solution L(ε) to (6.15) for ε ↓ 0. For this wetake the logarithm of (6.15) to obtain with y := log ε−2 (with y →∞ as ε ↓ 0).

L =y

c logM+

log (p+ qL)

c logM. (6.16)

By reformulating (6.16) as

L =y

c logM+

1

c logM

(logL+ log q + log

(p

qL+ 1

))=

y

c logM+

log q

c logM+

logL

c logM+O(L−1), y →∞, (6.17)

147


and then writing (6.17) as

L =

yc logM + log q

c logM +O(L−1)

1− logLLc logM

we easily observe that L = O(y) as y →∞. We thus get by iterating (6.16),

L =y

c logM+O(log y),

and iterating once again,

L =y

c logM+

log(p+ qy

c logM +O(log y))

c logM

=y

c logM+

log(p+ qy

c logM

)c logM

+O(log y

y).

The next few iterations yield

L =y

c logM+

log

(p+ qy

c logM +q log

(p+ qy

c logM

)c logM

)c logM

+O(log y

y2), and

L =y

c logM+

log

p+ qyc logM +

q log

(p+ qy

c logM+q log(p+ qy

c logM )c logM

)c logM

c logM

+O(log y

y3),

etc. After re-expressing the asymptotic solution in the origonal terms via

a =

√c2

c′2

M + 1

MT

b = c21T

2α

p = c21T

2α

(1 +

√c′2c2T

M

M + 1α logM

)q = c2

1αT2α logM

we gather, respectively,

L(ε) =log ε−1

α logM+O(log log ε−1),

L(ε) =1

α logMlog ε−1 (6.18)

+1

2α logMlog

c21T

2α

1 +

√c′2c2T

M

M + 1α logM

+ c21T

2α log ε−1

+O(

log log ε−1

log ε−1)

etc. The resulting complexity (6.9) will be obviously

Cβ=1(L) = O(log2 ε−1

ε2),

148

Multilevel Monte Carlo for RDEs

where sharper expressions can be obtained by inserting one of the above asymptotic expansionsfor L(ε). Note that (6.18) may be written as

L(ε) =1

α logMlog

ε−1

c21T

2α

1 +

√c′2c2T

M

M + 1α logM

+ c21T

2α log ε−1

1/2

+O(log log ε−1

log ε−1)

which suggest that in (6.10),

d1 ≈

1 +

√c′2c2T

M

M + 1α logM + log ε−1

1/2

for the case β = 1.

6.4 Multilevel Monte Carlo for RDEs

Let X : [0, T ] → Rd be Gaussian with the same assumptions as in Theorem 6.2.1 for someρ ∈ [1, 2). Consider the solution Y : [0, T ]→ Rm of the RDE

dYt = V (Yt) dXt; Y0 ∈ Rm

where V = (V1, . . . , Vd) is a collection of vector fields in Rm with |V |Lipγ <∞ for some γ ≥ 2ρρ−1 .

Set S := Y and let S(hl) be the simplified step-3 Euler approximation of Y with mesh-size hl(in the case ρ = 1, it suffices to consider a step-2 approximation). Let f : C([0, T ],Rm) → Rnbe a Lipschitz continuous functional and set P := f(S), Pl := f(S(hl)). We want to calculatethe quantities needed in Theorem 6.3.1. It suffices to apply the modified complexity theoremwith α = β/2. To wit,

V[Pl − P

]≤ E

[(Pl − P

)2]≤ |f |2LipE

[∣∣∣S(hl) − S∣∣∣2] = O

(hβl

)and

V[Pl − Pl−1

]≤(V[Pl − P

]1/2+ V

[Pl−1 − P

]1/2)2

= O(hβl

)for all β < 2

ρ − 1. Of course the variance of the average of Nl IID samples becomes

V[Yl

]=

1

NlV[Pl − Pl−1

]= O

(hβl /Nl

).

This shows (iii). Trivially, a strong rate is also a weak rate, in the sense that

E(Pl − P

)≤ E

[(Pl − P

)2]1/2

= O(hβ/2l

).

Condition (ii), “unbiasedness” is obvious for the estimator (6.6). Finally, the computationalcomplexity Yl is obviously bounded by O (Nl/hl). (Create Nl samples paths with step-size ∼hl). Corollaries 6.2.6 and 6.3.2 then imply

Theorem 6.4.1. The Monte Carlo evaluation of a functional of an RDE driven by Gaussiansignal, to within a MSE of ε2, can be achieved with computational complexity

O(ε−θ)∀θ > 2ρ

2− ρ.

149


150

Appendix

A Kolmogorov theorem for multiplicative functionals

The next Lemma is a slight modification of [FV10b, Theorem A.13]. The proof follows the ideasof [FH, Theorem 3.1].

Lemma A.1.1 (Kolmogorov for multiplicative functionals). Let X,Y : [0, T ] × Ω → TN (V )be random multiplicative functionals and assume that X(ω) and Y(ω) are continuous for allω ∈ Ω. Let β, δ ∈ (0, 1] and choose β′ < β and δ′ < δ. Assume that there is a constant M > 0such that

|Xns,t|Lq/n ≤M

n|t− s|nβ

|Yns,t|Lq/n ≤M

n|t− s|nβ

|Xns,t −Yn

s,t|Lq/n ≤Mnε|t− s|δ+(n−1)β

hold for all s < t ∈ [0, T ] and n = 1, . . . , N where ε is a positive constant and q ≥ q0 where

q0 := 1 +

(1

β − β′∨ 1

δ − δ′

).

Then there is a constant C = C(N, β, β′, δ, δ′) such that∣∣∣∣∣ sups<t∈[0,T ]

|Xns,t|

|t− s|nβ′

∣∣∣∣∣Lqn

≤ CMn (A.19)∣∣∣∣∣ sups<t∈[0,T ]

|Yns,t|

|t− s|nβ′

∣∣∣∣∣Lqn

≤ CMn (A.20)∣∣∣∣∣ sups<t∈[0,T ]

|Xns,t −Yn

s,t||t− s|δ′+(n−1)β′

∣∣∣∣∣Lqn

≤ CMnε (A.21)

hold for all n = 1, . . . , N .

Proof. W.l.o.g., we may assume T = 1. Let (Dk)k∈N be the sequence of dyadic partitions of theinterval [0, 1), i.e. Dk =

l

2k: l = 0, . . . , 2k − 1

. Clearly, |Dk| = 1

#Dk= 2−k. Set

Knk,X := max

ti∈Dk|Xn

ti,ti+1|

Knk,Y := max

ti∈Dk|Yn

ti,ti+1|

Knk,X−Y :=

1

εmaxti∈Dk

|Xnti,ti+1

−Ynti,ti+1

|

Kolmogorov multiplicative functionals

for n = 1, . . . , N and k ∈ N. By assumption, we have

E|Knk,X |

qn ≤ E

∑ti∈Dk

|Xnti,ti+1

|qn ≤ #Dk max

ti∈DkE|Xn

ti,ti+1|qn ≤M q|Dk|qβ−1.

In the same way one estimates Knk,Y and Kn

k,X−Y , hence

|Knk,X |Lq/n ≤M

n|Dk|nβ−n/q (A.22)

|Knk,Y |Lq/n ≤M

n|Dk|nβ−n/q (A.23)

|Knk,X−Y |Lq/n ≤M

n|Dk|δ+(n−1)β−n/q. (A.24)

Note the following fact: For any dyadic rationals s < t, i.e. s < t ∈ ∆ :=⋃∞k=1Dk, there is a

m ∈ N such that |Dm+1| < |t− s| ≤ |Dm| and a partition

s = τ0 < τ1 < . . . < τN = t (A.25)

of the interval [s, t) with the property that for any i = 0, . . . , N − 1 there is a k ≥ m + 1 with[τi, τi+1) ∈ Dk, but for fixed k ≥ m+ 1 there are at most two such intervals contained in Dk.

Step 1: We claim that for every n = 1, . . . , N there is a real random variable KnX such that

|KnX |Lq/n ≤Mnc where c = c(β, β′, δ, δ′) and that for any dyadic rationals s < t and m, (τi)

Ni=0

chosen as in (A.25) we have

N−1∑i=0

|Xnτi,τi+1

||t− s|nβ′

≤ KnX . (A.26)

Furthermore, the estimate (A.26) also holds for Yn and a random variable KnY . Indeed: By the

choice of m and (τi)Ni=0,

N−1∑i=0

|Xnτi,τi+1

||t− s|nβ′

≤∞∑

k=m+1

2Knk,X

|Dm+1|nβ′≤ 2

∞∑k=m+1

Knk,X

|Dk|nβ′≤ 2

∞∑k=1

Knk,X

|Dk|nβ′=: Kn

X .

It remains to prove that |KnX |Lq/n ≤Mnc. By the triangle inequality and the estimate (A.22),∣∣∣∣∣

∞∑k=1

Knk,X

|Dk|nβ′

∣∣∣∣∣Lq/n

≤Mn∞∑k=1

|Dk|n(β−1/q−β′) ≤Mn∞∑k=1

|Dk|(β−1/q0−β′) <∞

since β − 1/q0 − β′ > 0 which shows the claim.

Step 2: We show that (A.19) and (A.20) hold for all n = 1, . . . , N . It is enough to consider

X. Note first that, due to continuity, it is enough to show the estimate for sups<t∈∆|Xns,t|

|t−s|nβ′ . By

induction over n: For n = 1, this just follows from the usual Kolmogorov continuity criterion.Assume that the estimate is proven up to level n − 1. Let s < t be any dyadic rationals andchoose m and (τi)

Ni=0 as in (A.25). Since X is a multiplicative functional,

|Xns,t| ≤

N−1∑i=0

|Xnτi,τi+1

|+n−1∑l=1

maxi=1,...,N

|Xn−ls,τi |

N−1∑i=0

|Xlτi,τi+1

|

and thus, using step 1,

|Xns,t|

|t− s|nβ′≤

N−1∑i=0

|Xnτi,τi+1

||t− s|nβ′

+

n−1∑l=1

supu<v∈∆

|Xn−lu,v |

|v − u|(n−l)β′N−1∑i=0

|Xlτi,τi+1

||t− s|lβ′

≤ KnX +

n−1∑l=1

supu<v∈∆

|Xn−lu,v |

|v − u|(n−l)β′K lX .

152

We can now take the supremum over all s < t ∈ ∆ on the left. Taking the Lq/n-norm on bothsides, using first the triangle, then the Holder inequality and the estimates from step 1 togetherwith the induction hypothesis gives the claim.

Step 3: As in step 1, we claim that for any n = 1, . . . , N there is a random variableKnX−Y ∈ Lq/n such that for any dyadic rationals s < t and m, (τi)

Ni=0 chosen as above we have

N−1∑i=0

|Xnτi,τi+1

−Ynτi,τi+1

||t− s|δ′+(n−1)β′

≤ KnX−Y ε. (A.27)

Furthermore, we claim that |KnX−Y |Lq/n ≤ Mnc where c = c(β, β′, δ, δ′). The proof follows the

lines of step 1, setting

1

ε

N−1∑i=0

|Xnτi,τi+1

−Ynτi,τi+1

||t− s|δ′+(n−1)β′

≤ 2

∞∑k=1

Knk,X−Y

|Dk|δ′+(n−1)β′=: Kn

X−Y .

Step 4: We prove that (A.21) holds for all n = 1, . . . , N . By induction over n: The casen = 1 is again just the usual Kolmogorov continuity criterion applied to t 7→ ε−1(Xt − Yt).Assume the assertion is shown up to level n − 1 and chose two dyadic rationals s < t. Usingthe multiplicative property, we have

|Xns,t −Yn

s,t| ≤N−1∑i=0

|Xnτi,τi+1

−Ynτi,τi+1

|+n−1∑l=1

maxi=1,...,N

|Xn−ls,τi |

N−1∑i=0

|Xlτi,τi+1

−Ylτi,τi+1

|

+

n−1∑l=1

maxi=1,...,N

|Xn−ls,τi −Yn−l

s,τi |N−1∑i=0

|Ylτi,τi+1

|.

Now we proceed as in step 2, using the estimates from step 1 to step 3 and the inductionhypothesis.

153

Kolmogorov multiplicative functionals

154

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