on parabolic stochastic integro-differential …on parabolic stochastic integro-differential...

On Parabolic StochasticIntegro-Differential Equations

Existence, Regularity, and Numerics

James-Michael Leahy

Doctor of PhilosophyUniversity of Edinburgh

June 2015

Declaration

I hereby declare that this thesis was composed by myself and that the work containedherein is my own, except where explicitly stated otherwise. Further, I declare that thiswork has not been submitted for any other degree or professional qualification.

James-Michael LeahyJune 2015

i

Abstract

In this thesis, we study the existence, uniqueness, and regularity of systems of degener-ate linear stochastic integro-differential equations (SIDEs) of parabolic type with adaptedcoefficients in the whole space. We also investigate explicit and implicit finite differenceschemes for SIDEs with non-degenerate diffusion. The class of equations we considerarise in non-linear filtering of semimartingales with jumps.

In Chapter 2, we derive moment estimates and a strong limit theorem for space inversesof stochastic flows generated by Lévy driven stochastic differential equations (SDEs) withadapted coefficients in weighted Hölder norms using the Sobolev embedding theorem andthe change of variable formula. As an application of some basic properties of flows ofWeiner driven SDEs, we prove the existence and uniqueness of classical solutions of lin-ear parabolic second order stochastic partial differential equations (SPDEs) by partition-ing the time interval and passing to the limit. The methods we use allow us to improveon previously known results in the continuous case and to derive new ones in the jumpcase. Chapter 3 is dedicated to the proof of existence and uniqueness of classical solutionsof degenerate SIDEs using the method of stochastic characteristics. More precisely, weuse Feynman-Kac transformations, conditioning, and the interlacing of space inverses ofstochastic flows generated by SDEs with jumps to construct solutions.

In Chapter 4, we prove the existence and uniqueness of solutions of degenerate linearstochastic evolution equations driven by jump processes in a Hilbert scale using the varia-tional framework of stochastic evolution equations and the method of vanishing viscosity.As an application, we establish the existence and uniqueness of solutions of degeneratelinear stochastic integro-differential equations in the L2-Sobolev scale.

Finite difference schemes for non-degenerate SIDEs are considered in Chapter 5.Specifically, we study the rate of convergence of an explicit and an implicit-explicit finitedifference scheme for linear SIDEs and show that the rate is of order one in space andorder one-half in time.

Key words: Stochastic flows, stochastic differential equations (SDEs), Lévy processes, strong-limit the-

orem, stochastic partial differential equations (SPDEs), degenerate parabolic type, parabolic stochastic

integro-differential equations (SIDEs), partial integro-differential equations (PIDEs), non-local operators,

method of stochastic characteristics, Ito-Wentzell formula, stochastic evolution equations, vanishing viscos-

ity, finite difference schemes

AMS subject classifications (MSC2010). 60H10, 60J75, 60F15, 60H15, 34F05, 35F40, 35R60, 35K15,

35K65, 60H20, 45K05, 65M06, 65M12

ii

Lay Summary

In this thesis, we investigate a class of equations governing random processes that dependon time and space. These equations are a generalization of deterministic parabolic partialdifferential equations (PDEs). The prototypical example of a parabolic PDE is the heatequation, which is an equation that governs the evolution of heat in some medium in timeand space. Randomness can be introduced to the heat equation through a random sourceand sink of heat at certain points of time and space, a random initial heat profile, anda random conductivity coefficient of the medium. The heat equation is an example ofa diffusion equation, which is an equation that describes the density of particles as theydiffuse according to some law. The physical law that governs the heat equation is calledFick’s law and it describes only a special type of diffusion. The equations that we study,even if we take them to be deterministic, allow for a much wider range of diffusions thanthe prototypical heat equation.

When investigating an equation, a natural first question to ask is whether there existsa solution to the equation in some well-defined sense. Moreover, if there exists a solution,then one ought to ask whether it is unique, and if so, what properties it has. The equationswe consider have range of possible inputs, and thus we want to know how the properties ofthe solution depend on the inputs. There are many properties of the equations to explore,but we consider only a property that is referred to as regularity. Regularity is characterizedby a class of spaces to which the solution and inputs belong that are ordered by inclusion;smaller spaces correspond to higher regularity. It is natural to expect that if you take moreregular inputs, then the solution should be more regular. This is the case for the equationswe consider.

It is rare for the solution of an equation we study to have a closed form expression oftime, space, and randomness. Nevertheless, it is still possible to prove that there existsa unique solution and to study the regularity of that solution. In fact, there are differentapproaches to accomplish this objective and each way has its own advantage. We will dis-cuss two such approaches in this thesis. Since we can not expect a closed form expressionfor the solutions, it also practical and interesting to develop some method to approximatethe solutions and to prove that the approximate solutions are close in some well-definedsense to the true solution that we know exists. In the final chapter, we describe a simpleapproximation scheme for a special subclass of the equations considered. The approxi-mate solutions are defined on some countable set of points in time and space and satisfyan equation that can be solved by simple algebraic manipulations for some realization ofthe random variables in the equation.

iii

I dedicate this thesis to my family.

iv

Acknowledgements

First, I would like to thank my supervisor, István Gyöngy, for the support and guidancehe has provided me during my time at the University of Edinburgh. I am indebted toRemigijus Mikulevicius for the opportunity to work with him at the University of SouthernCalifornia in the Fall of 2013 and to collaborate with him on numerous projects. I havealso benefited from having Kontantinos Dareiotis as a friend, flatmate, and collaborator.

The research contained in this thesis would not have been possible without fundingfrom the Principal’s Career Development Scholarship and the Edinburgh Global ResearchScholarship and support from Gill Law, Graduate School Administrator.

I have been fortunate enough to participate in numerous discussions with other studentsthat have directly contributed to my academic development. In particular, I would like tomention: Eric Hall, Máté Gerencsèr, Brian Hamilton, Marc Ryser, David Cottrell, AndreaMeireles Rodrigues, Sebastian Vollmer, Chaman Kumar, and José Rodríguez Villarreal.In fact, it was Marc Ryser who introduced me to Stochastic Partial Differential Equationsand suggested I apply to the University of Edinburgh to work with István Gyöngy.

I would like to thank my friends and family, especially my mother, grandparents, andAunt Mary Ann and Uncle Greg, for supporting all my endeavours. Last, I am grateful forthe love, patience, and encouragement Bihotz Barrenetxea Dominguez has shown me.

v

Contents

Declaration i

Abstract ii

Lay Summary iii

Acknowledgements v

1 Introduction 1

2 Properties of space inverses of stochastic flows 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Properties of stochastic flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Homeomorphism property of flows . . . . . . . . . . . . . . . . . . . 132.3.2 Moment estimates of inverse flows: Proof of Theorem 2.2.1 . . . . . 172.3.3 Strong limit of a sequence of flows: Proof of Theorem 2.2.3 . . . . . 22

2.4 Classical solutions of degenerate SPDEs: Proof of Theorem 2.2.4 . . . . . . 262.5 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 The method of stochastic characteristics for parabolic SIDEs 373.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3 Proof of main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.1 Proof of uniqueness for Theorem 3.2.2 . . . . . . . . . . . . . . . . . 463.3.2 Small jump case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.3 Adding free and zero-order terms. . . . . . . . . . . . . . . . . . . . . 553.3.4 Adding uncorrelated part (Proof of Theorem 3.2.2) . . . . . . . . . . 613.3.5 Proof of Theorem 3.2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4 Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.4.1 Martingale and point measure moment estimates . . . . . . . . . . . 673.4.2 Optional projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4.3 Estimates of Hölder continuous functions . . . . . . . . . . . . . . . . 693.4.4 Stochastic Fubini theorem . . . . . . . . . . . . . . . . . . . . . . . . . 803.4.5 Itô-Wentzell formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

vi

4 The L2-Sobolev theory for parabolic SIDEs 874.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2 Degenerate linear stochastic evolution equations . . . . . . . . . . . . . . . . 89

4.2.1 Basic notation and definitions . . . . . . . . . . . . . . . . . . . . . . . 894.2.2 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . 934.2.3 Proof of Theorem 4.2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.3 The L2-Sobolev theory for degenerate SIDEs . . . . . . . . . . . . . . . . . . 1044.3.1 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . 1044.3.2 Proof of Theorem 4.3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.3.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5 A finite difference scheme for non-degenerate parabolic SIDEs 1285.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.2 Statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.4 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.5 Proof of the main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6 Conclusions and future work 162

References 163

vii

Chapter 1IntroductionThe subject of this thesis is parabolic linear stochastic integro-differential equations (SIDEs).These equations arise in the study of non-linear filtering of semimartingales, scaled lim-its of particle systems, flows of stochastic diffeomorphisms, and mathematical biology,physics, and finance. For the author, the strongest impetus to the advancement of thetheory of SIDEs is the problem of non-linear filtering of jump diffusions.

The first derivation of the filtering equations for semimartingales with jumps is due toB. Grigelionis in [Gri72]. The reduced form filtering equations, which are linear SIDEs,were derived in [Gri76]. A proof of existence and uniqueness of solutions to the reducedform filtering equations was given in [Tin77b] by E. Tinfavicius under the assumption ofnon-degenerate stochastic parabolicity. While in [Tin77b] essentially the variational ap-proach of SPDEs was used, one important aspect of this approach was missing, namelyItô’s formula for the square of the norm for jump processes. Such a result is used toconclude that the variational solution of a stochastic evolution equation is càdlàg withvalues in the pivot space and to get an energy estimate of the supremum in time of thenorm of the solution in the pivot space. In [GK81], I. Gyöngy and N.V. Krylov derivedItô’s formula for the square of the norm in the jump case, and the corresponding exis-tence and uniqueness result for monotone stochastic evolution equations with jumps wasstudied in [Gyö82]. In [Gri82], B. Grigelionis applied the result in [GK81] to completethe variational existence and uniqueness result for the reduced form equations in the non-degenerate setting that began in [Tin77b]. The reader that is interested in the derivation ofthe reduced form filtering equations for semimartingales with jumps is advised to consultthe article [Gri82] and the review article [GM11].

It is also worth mentioning that the stochastic evolution equations driven by addi-tive càdlàg martingale noise were studied by G. Pistone and M. Métivier in Section 5 of[MP76] using the semigroup approach. One important aspect of the reduced form filter-ing equation is that, in general, the principal part of the operators in the drift of theseequations depend on space and time and are random. The standard semigroup approach tostochastic equations in infinite dimensions cannot treat these type of equations, since if thesemigroup generated by the principal part is random, the stochastic convolution does notmake sense as an ordinary Itô integral. In a recent work, M. Pronk and M. Veraar [PV14]showed that it is possible to extend the semigroup approach to treat equations where the

1

2

principal part of stochastic evolution equation is random. However, at the time of writing,there still seems to be some limitations in this approach; for example, some regularity intime is needed for the diffusion coefficient in the stochastic heat equation when applyingtheir theory. We emphasize that this is not needed in the variational approach.

This thesis is dedicated to the the proof of existence, uniqueness, and regularity offully degenerate linear SIDEs. Specifically, we derive a theory for these equations inHölder spaces using the method of stochastic characteristics and a theory in L2-Sobolevspaces using the variational approach of stochastic evolution equations and the method ofvanishing viscosity. We also investigate finite difference approximations of SIDEs withnon-degenerate diffusion.

Let us state the general form of the equation that we will investigate in this thesis. Let(Ω,F ,F = (Ft)t≥0,P) be a complete filtered probability space satisfying the usual condi-tions of right-continuity and completeness. For each real number T > 0, we let RT and PT

be the F-progressive and F-predictable sigma-algebra on Ω × [0,T ], respectively. For ourdriving processes, we take a sequence wϱ

t , t ≥ 0, ϱ ∈ N, of independent one-dimensional F-adapted Wiener processes and a F-adapted Poisson random measure p(dt, dz) on (R+×Z1,

B(R+)⊗Z1) with intensity measure π1(dz)dt, where (Z1,Z1, π1) is a sigma-finite measurespace. Denote by q(dt, dz) = p(dt, dz) − π1(dz)dt the compensated Poisson random mea-sure. Let D1, E1,V1 ∈ Z be disjoint Z1-measurable subsets such that D1 ∪ E1 ∪ V1 = Z1

and π(V1) < ∞. Let (Z2,Z2, π2) be a sigma-finite measure space and D2, E2 ∈ Z2 be dis-jointZ2-measurable subsets such that D2 ∪ E2 = Z2.

Fix an arbitrary positive real number T > 0 and integers d1, d2 ≥ 1. Let α ∈ (0, 2] andlet φ : Ω × Rd1 → Rd2 be F0 ⊗ B(Rd1)-measurable. We consider the system of SIDEs on[0,T ] × Rd1 given by

dult =

((L1;l

t +L2;lt )ut + 1[1,2](α)bi;ll

t ∂iult + cll

t ult + f l

t

)dt +

(N

lϱt ut + glϱ

t

)dwϱ

t

+

∫Z1

(I

1;lt,z ut− + hl

t(z))

[1D1(z)q(dt, dz) + 1E1∪V1(z)p(dt, dz)], t ≤ T,

ul0 = φ

l, l ∈ 1, . . . , d2, SIDE

where for ϕ ∈ C∞c (Rd1; Rd2), k ∈ 1, 2, and l ∈ 1, . . . , d2,

Lk;lt ϕ(x) : = 12(α)ak;i j

t (x)∂i jϕl(x) +

∫Dkρk;ll

t (x, z)(ϕl(x + Hk

t (x, z)) − ϕl(x))πk(dz)

+

∫Dk

(ϕl(x + Hk

t (x, z)) − ϕl(x) − 1(1,2](α)Hk;it (x, z)∂iϕ

l(x))πk(dz)

+ 12(k)∫

E2

((Ill

d2+ ρ2;ll

t (x, z))ϕl(x + H2t (x, z)) − ϕl(x)

)π2(dz),

Nlϱt ϕ(x) : = 12(α)σiϱ

t (x)∂iϕl(x) + υllϱ

t (x)ϕl(x), ϱ ∈ N,

3

Ilt,zϕ(x) : = (Ill

d2+ ρ1;ll

t (x, z))ϕl(x + H1t (x, z)) − ϕl(x),

and ∫Dk

(|Hk

t (x, z)|α + |ρkt (x, z)|2

)πk(dz) +

∫Ek

(|Hk

t (x, z)|1∧α + |ρkt (x, z)|

)πk(dz) < ∞.

The summation convention with respect to repeated indices i, j ∈ 1, . . . , d1, l ∈ 1, . . . ,d2, and ϱ ∈ N is used here and below. The d2 × d2 dimensional identity matrix is denotedby Id2 . For a subset A of a larger set X, 1A denotes the 0, 1-valued function taking thevalue 1 on the set A and 0 on the complement of A. We assume that for each k ∈ 1, 2,

σkt (x) = (σk;iϱ

t (ω, x))1≤i≤d1, ϱ∈N, bt(x) = (bit(ω, x))1≤i≤d1 , ct(x) = (cll

t (ω, x))1≤l,l≤d2 ,

υkt (x) = (υk;llϱ

t (ω, x))1≤l,l≤d2, ϱ∈N, ft(x) = ( f it (ω, x))1≤i≤d2 , gt(x) = (giϱ

t (ω, x))1≤i≤d2, ϱ∈N,

are random fields on Ω× [0,T ]×Rd1 that are RT ⊗B(Rd1)-measurable. For each k ∈ 1, 2,we assume that

Hkt (x, z) = (Hk;i

t (ω, x, z))1≤i≤d1 , ρkt (x, z) = (ρk;ll

t (ω, x, z))1≤l,l≤d2 ,

are random fields onΩ×[0,T ]×Rd1×Zk that are PT ⊗B(Rd1)⊗Zk-measurable. Moreover,we assume that ht(x, z) = (hi

t(ω, x, z))1≤i≤d2 is a random field on Ω × [0,T ] × Rd1 × Z1 thatis PT ⊗ B(Rd1) ⊗Z1-measurable.

This thesis is organized as follows. Chapter 2 is dedicated to establishing some prop-erties of space inverses of stochastic flows generated by an SDEs with jumps. Theseproperties play an important role in Chapter 3 when we construct classical solutions of(SIDE). Specifically, in Chapter 2, we derive moment estimates and a strong limit theoremfor space inverses of stochastic flows generated by jump SDEs with adapted coefficients inweighted Hölder norms. We also give a relatively simple, but novel, proof of existence anduniqueness of classical solutions of a degenerate SPDE (i.e. (SIDE) with a1;i j = σiϱσ jϱ,a2 ≡ H1 ≡ H2 ≡ ρ1 ≡ ρ2 ≡ b ≡ c ≡ f ≡ υ ≡ g ≡ h ≡ 0, and φ(x) = x). Our method ofproof allows us to prove existence and uniqueness when σ can degenerate on set of pos-itive probability under less regularity than done previously. If σ is non-degenerate, thenmuch more can be done, and we refer the reader to the seminal work [FGP10].

The focus of Chapter 3 is to give a complete proof of existence and uniqueness of clas-sical solutions of (SIDE) with ai j = σiϱσ jϱ + σ2;iϱσ2; jϱ and with Hölder assumptions onthe coefficients and data using Feynman-Kac transformations, conditioning, and the inter-lacing of space-inverses of stochastic flows associated with the equations. We emphasizethat σ,σ2,H, and H2 can vanish on a set of positive probability.

In Chapter 4, we prove the existence and uniqueness of solutions of degenerate linear

4

stochastic evolution equations driven by jump processes in a Hilbert scale using the varia-tional framework of stochastic evolution equations and the method of vanishing viscosity.This result extends the work of B. Rozovskii in [Roz90] from the continuous case to thejump case. As an application of our result, we derive unique solutions of (SIDE) witha1;i j ≡ a2;i j ≡ σ ≡ g ≡ 0, E2 = E1 = V1 = ∅, and α ∈ (1, 2) in the integer L2-Sobolev scale.We confine ourself to an equation without continuous noise and diffusion because the the-ory of Sobolev solutions for equations in this case are well-studied (see, e.g. [KR82],[Roz90], and [GGK14]). The results derived in Chapters 1, 2, and 3 are the fruit of acollaborative effort with Remigijus Mikulevicius at the University of Southern California;both R. Mikulevicius and the author have contributed substantially to the development ofthe ideas contained in these chapters.

Finally, in Chapter 5, we consider finite difference schemes for (SIDE) with d2 = 1,H1

t (x, z) = z, ρ1 = 1, E1 = V1 = ∅, α = 2, h ≡ H2 ≡ ρ2 ≡ 0, and ai j − σiϱσ jϱ > 0.We show that the L2(Ω)-pointwise rate of convergence is of order one in space and orderone-half in time. The results given in this chapter are the fruit of a collaborative effort withKonstantinos Dareiotis at the University of Edinburgh; both K. Dareiotis and the authorhave contributed substantially to the development of the ideas contained in this chapter.

Basic Notation

Let N be the set of natural numbers and N0 = 0, 1, . . . be the set of natural numbersincluding zero. Let Z be the set of integers. For a topological space (X,X), we denote theBorel sigma-field on X by B(X).

For each integer n ≥ 1, let Rn be the n-dimensional Euclidean space and for eachx ∈ Rn, denote by |x| the Euclidean norm of x = (x1, . . . , xn). We set R = R1 and fortwo elements of x, y ∈ R, we denote by x ∨ y the maximum of x and y and by x ∧ y, theminimum of x and y. Let R+ denote the set of non-negative elements of R1.

For an integer n ≥ 1 and for each i ∈ 1, . . . , n, we let ∂i =∂∂xi

be the spatial derivativeoperator with respect to the direction xi and write ∂i j = ∂i∂ j for each i, j ∈ 1, . . . , n. Wealso denote the identity operator by ∂0. For a once differentiable function f = ( f 1 . . . , f n) :Rn → Rn, we denote the gradient of f by ∇ f = (∂ j f i)1≤i, j≤d1 . For a multi-index γ =

(γ1, . . . , γd) ∈ 0, 1, 2, . . . , d1 of length |γ| := γ1 + · · · + γd, denote by ∂γ the operator∂γ = ∂

γ11 · · · ∂

γdd , where ∂0

i is the identity operator for all i ∈ 1, . . . , d1. For i ∈ 1, . . . , d,let ∂−i = −∂i.

For integers d1, d2 ≥ 1, we denote by C∞c (Rd1; Rd2) the space of Rd2-valued infinitelydifferentiable functions with compact support in Rd1 .

For each integer n ≥ 1, the norm of an element x of ℓ2(Rn), the space of square-summable Rn-valued sequences, is denoted by |x|. We set ℓ2 = ℓ2(R). For integers n ≥ 1

5

and a once differentiable function f = ( f 1ϱ, . . . , f nϱ)ϱ∈N : Rn → ℓ2(Rn), we denote thegradient of f by ∇ f = (∂ j f iϱ)1≤i, j≤n,ϱ∈N and understand it as a function from Rd1 to ℓ2(Rn2

).For a Fréchet space χ and fixed time T > 0, we denote by D([0,T ]; χ) the space of

χ-valued càdlàg (continuous from the right and limits from the left) functions on [0,T ]and by C([0,T ]2; χ) the space of χ-valued continuous functions on [0,T ] × [0,T ]. Thespaces D([0,T ]; χ) and C([0,T ]2; χ) are Fréchet spaces when endowed with the set of thesupremum (in time) seminorms, which we assume, unless otherwise specified.

The notation N = N(·, · · · , ·) is used to denote a positive constant depending only onthe quantities appearing in the parentheses. In a given context, the same letter is oftenused to denote different constants depending on the same parameter. Moreover, for anyfunction f defined on a set X and taking valued in a linear space Y with zero element 0Y ,the notation f ≡ 0 indicates that f (x) = 0Y for all x ∈ X.

Chapter 2Properties of space inverses of stochasticflows

2.1 Introduction

Let (Ω,F ,F = (Ft)t≥0,P) be a complete filtered probability space satisfying the usual con-ditions of right-continuity and completeness. Let wϱ

t , t ≥ 0, ϱ ∈ N, be a sequence of inde-pendent one-dimensional F-adapted Wiener processes. For a sigma-finite measure space(Z,Z, π), we let p(dt, dz) be an F-adapted Poisson random measure on (R+×Z,B(R+)⊗Z)with intensity measure π(dz)dt and denote by q(dt, dz) = p(dt, dz) − π(dz)dt the compen-sated Poisson random measure. For each real number T > 0, we let RT and PT be theF-progressive and F-predictable sigma-algebra on Ω × [0,T ], respectively.

Fix a real number T > 0 and an integer d ≥ 1. For each stopping time τ ≤ T , considerthe flow Xt = Xt(τ, x), (t, x) ∈ [0,T ] × Rd, generated by the SDE

dXt = bt(Xt)dt + σϱt (Xt)dwϱ

t +

∫Z

Ht(Xt−, z)q(dt, dz), τ < t ≤ T,

Xt = x, t ≤ τ, (2.1.1)

where bt(x) = (bit(ω, x))1≤i≤d and σt(x) = (σiϱ

t (ω, x)1≤i≤d,ϱ∈N are Rd-valued RT ⊗ B(Rd)-measurable functions defined on Ω × [0,T ] × Rd and Ht(x, z) = (Hi

t(ω, x, z))1≤i≤d is anRd-valued PT ⊗ B(Rd) ⊗ Z-measurable function defined on Ω × [0,T ] × Rd × Z. Thesummation convention with respect to the repeated index ϱ is used here and below.

In this chapter, under Hölder regularity conditions on the coefficients b, σ, and H, weprovide a simple and direct derivation of moment estimates of the space inverse of theflow, denoted X−1

t (τ, x), in weighted Hölder norms. This is done by applying the Sobolevembedding theorem and the change of variable formula. Using a similar method, we es-tablish a strong limit theorem in weighted Hölder norms for a sequence of flows X(n)

t (τ, x)and their inverses X(n);−1

t (τ, x) corresponding to a sequence of coefficients (b(n), σ(n),H(n))converging in weighted Hölder norms. Furthermore, as an application of the diffeomor-phism property of flow, we give a direct derivation of the linear second order degenerateSPDE governing the inverse flow X−1

t (τ, x) when H ≡ 0. Specifically, for each τ ≤ T ,

6

2.1. Introduction 7

consider the stochastic flow Yt = Yt(τ, x), (t, x) ∈ [0,T ] × Rd, generated by the SDE

dYt = bt(Yt)dt + σϱt (Yt)dwϱ

t , τ < t ≤ T,

Yt = x, t ≤ τ.

Assume that b and σ have linear growth, bounded first and second derivatives, and thatthe second derivatives of b and σ are α-Hölder for some α > 0. By partitioning the timeinterval and using Taylor’s theorem, the Sobolev embedding theorem, and some basicproperties of the flow and its inverse, we show that ut(x) = ut(τ, x) := Y−1

t (τ, x), (t, x) ∈[0,T ] × Rd is the unique classical solution of the SPDE given by

dut(x) =(12σ

iϱt (x)σ jϱ

t (x)∂i jut(x) − bit(x)∂iut(x)

)dt − σiϱ

t (x)∂iut(x)dwϱt , τ < t ≤ T,

ut(x) = x, t ≤ τ, (2.1.2)

wherebi

t(x) = bit(x) − σ jϱ

t (x)∂ jσiϱt (x).

In Chapter 3, we will use all of the properties of the flow Xt(τ, x) that are established inthis first chapter in order to derive the existence and uniqueness of classical solutions oflinear parabolic SIDEs.

One of the earliest works to investigate the homeomorphism property of flowsof SDEs with jumps is by P. Meyer in [Mey81]. In [Mik83], R. Mikulevicius extendedthe properties found in [Mey81] to SDEs driven by arbitrary continuous martingales andrandom measures. Many other authors have since expanded upon the work in [Mey81],see for example [FK85, Kun04, MB07, QZ08, Zha13, Pri14] and references therein. In[Kun04, Kun86b], H. Kunita studied the diffeomorphism property of the flow Xt(s, x),(s, t, x) ∈ [0,T ]2 × Rd, and in the setting of deterministic coefficients, he showed that foreach fixed t, the inverse flow X−1

t (s, x), (s, x) ∈ [t,T ] × Rd, solves a backward SDE. Byestimating the associated backward SDE, one can obtain moment estimates and a stronglimit theorem for the inverse flow in essentially the same way that moment estimates areobtained for the direct flow (see, e.g. [Kun86b]). However, this method of deriving mo-ment estimates and a strong limit theorem for the inverse flow uses a time reversal, andthus ,requires that the coefficients are deterministic. In the case H ≡ 0, numerous authorshave investigated properties of the inverse flow with random coefficients. In Chapter 2 of[Bis81], Lemma 2.1 and 2.2 of [OP89], and Section 6.1 and 6.2 of [Kun96], propertiesof Y−1

t (τ, x) (i.e. moment estimates, strong limit theorem, and that it solves (2.1.2)) areestablished by first showing that the inverse flow solves the Stratonovich form SDE for

2.1. Introduction 8

Zt = Zt(τ, x), (t, x) ∈ [0,T ] × Rd, given by

dZt(x) = −Ut(Zt(x))bt(x)dt − Ut(Zt(x))σϱt (x) dwϱ

t , τ < t ≤ T, (2.1.3)

Z0(x) = x, τ < t,

where Ut(x) = Ut(τ, x) = ∇Yt(τ, x)−1. In order to obtain a strong solution of (2.1.3), condi-tions are imposed to ensure that ∇Ut(x) is locally-Lipschitz in x. In the degenerate setting,the third derivative of bt and σt need to be α-Hölder for some α > 0 to ensure that ∇Ut(x)is locally-Lipschitz in x. In direct contrast to this approach, we first derive properties ofthe inverse flow under the very assumptions that guarantee Yt(τ, x) is a diffeomorphism(i.e. when the first derivative of the coefficients are α-Hölder for some α > 0), and thenwe derive the existence of the equation without resorting to the SDE interpretation of theSPDE.

Classical solutions of (2.1.2) have been constructed in [Bis81, Kun96] by directlyshowing that Y−1

t (τ, x) solves (2.1.3). As we have mentioned above, this approach requiresthe third derivatives of bt and σt to be α-Hölder for some α > 0. Yet another approachto deriving existence of classical solutions of (2.1.2) is using the method of time reversal(see, e.g. [Kun96, DPT98]). While this method only requires that the second derivativesof bt and σt are α-Hölder for some α > 0, it does impose that the coefficients are deter-ministic. In [KR82], N.V. Krylov and B.L. Rozvskii derived the existence and uniquenessof generalized solutions of degenerate second order linear parabolic SPDEs in Sobolevspaces using variational approach of SPDEs and the method of vanishing viscosity (see,also, [GGK14] and Chapter 4, Section 2, Theorem 1 in [Roz90]). Thus, by appealing tothe Sobolev embedding theorem, this theory can be used to obtain classical solutions ofdegenerate linear SPDEs. Proposition 1 of Chapter 5, Section 2, in [Roz90] shows thatif σ is uniformly bounded and four-times continuously differentiable in x with uniformlybounded derivatives and b is uniformly bounded and three-times continuously differen-tiable with uniformly bounded derivatives, then there exists a classical solution of (2.1.2)and ut(x) = Y−1

t (x). This is more regularity than we require and we are also able to obtainsolutions in the entire Hölder scale.

This chapter is organized as follows. In Section 2.2, we state our notation and mainresults. Section 2.3 is devoted to deriving moment estimates and a strong limit theorem forthe space inverse of a stochastic flow generated by a Lévy driven SDE. In Section 2.4, weshow that Y−1

t (τ, x) is the unique classical solution of (2.1.2). In Section 2.5, the appendix,auxiliary facts that are used throughout the chapter are discussed.

2.2. Statement of main results 9

2.2 Statement of main results

Let us describe some notation that will be used in this chapter. Elements of Rd are un-derstood as column vectors and elements of Rd2

are understood as matrices of dimensiond × d. We denote the transpose of an element x ∈ Rd by x∗. For a Banach space V withnorm | · |V , domain (i.e. open connected set) Q of Rd, and continuous function f : Q→ V ,we define

| f |0;Q;V = supx∈Q| f (x)|V

and[ f ]β;Q;V = sup

x,y∈Q,x,y

| f (x) − f (y)|V|x − y|βV

, β ∈ (0, 1].

For any real number β ∈ R, we write β = [β]− + β+, where [β]− is an integer andβ+ ∈ (0, 1]. For a Banach space V with norm | · |V , real number β > 0, and domainQ of Rd, we denote by Cβ(Q; V) the Banach space of all bounded continuous functionsf : Q→ V having finite norm

| f |β;Q;V :=∑|γ|≤[β]−

|∂γ f |0;Q;V +∑|γ|=[β]−

[∂γ f ]β+;Q;V .

When Q = Rd and V = Rn or V = ℓ2(Rn) for any integer n ≥ 1, we drop the subscriptsQ and V from the norm | · |β;Q;V and write | · |β. For a Banach space V and for each β > 0,denote by Cβloc(R

d; V) the Fréchet space of continuous functions f : Rd → V satisfyingf ∈ Cβ(Q; V) for all bounded domains Q ⊂ Rd. We call a function f : Rd → Rd aCβloc(R

d; Rd)-diffeomorphism if f is a homeomorphism and both f and its inverse f −1 arein Cβloc(R

d; Rd).

If we do not specify to which space the parameters ω, t, x, y, z and n belong, then wemean ω ∈ Ω, t ∈ [0,T ], x, y ∈ Rd, z ∈ Z, and n ∈ N.

Let r1(x) =√

1 + |x|2, x ∈ Rd. For each real number β > 1, we introduce the followingregularity condition on the coefficients b, σ, and H.

Assumption 2.2.1 (β). (1) There is a constant N0 > 0 such that for all (ω, t, z) ∈ Ω ×[0,T ] × Z,

|r−11 bt|0 + |∇bt|β−1 + |r−1

1 σt|0 + |∇σt|β−1 ≤ N0 and |r−11 Ht(z)|0 + |∇Ht(z)|β−1 ≤ Kt(z),


where K : Ω × [0,T ] × Z → R+ is a PT ⊗Z-measurable function satisfying

Kt(z) +∫

ZKt(z)2π(dz) ≤ N0,

for all (ω, t, z) ∈ Ω × [0,T ] × Z.(2) There are constants η ∈ (0, 1) and Nκ > 0 such that for all (ω, t, x, z) ∈ (ω, t, x, z) ∈Ω × [0,T ] × Rd × Z : |∇Ht(ω, x, z)| > η,

| (Id + ∇Ht(x, z))−1| ≤ Nκ.

The following theorem shows that if Assumption 2.2.1 (β) holds for some β > 1, thenfor any β′ ∈ [1, β], the solution Xt(τ, x) of (2.1.1) has a modification that is a Cβ

′

loc(RdRd)

-diffeomorphism and the p-th moments of the weighted β′-Hölder norms of the inverseflow are bounded. This theorem will be proved in the next section.

Theorem 2.2.1. Let Assumption 2.2.1(β) hold for some β > 1.

(1) For any stopping time τ ≤ T and β′ ∈ [1, β), there exists a modification of the strongsolution Xt(τ, x) of (2.1.1), also denoted by Xt(τ, x), such that P-a.s. the mappingXt(τ, ·) : Rd → Rd is a Cβ

′

loc(Rd; Rd)-diffeomorphism, X·(τ, ·), X−1

· (τ, ·) ∈ D([0,T ];Cβ′

loc(Rd; Rd)), and X−1

t− (τ, ·) coincides with the inverse of Xt−(τ, ·). Moreover, for allϵ > 0 and p ≥ 2, there is a constant N = N(d, p,N0,T, β′, ϵ) such that

E[supt≤T|r−(1+ϵ)

1 Xt(τ)|p0

]+ E

[supt≤T|r−ϵ1 ∇Xt(τ)|pβ′−1

]≤ N

and a constant N = N(d, p,N0,T, β′, η,Nκ, ϵ) such that


1 X−1t (τ)|p0

]+ E

[supt≤T|r−ϵ1 ∇X−1

t (τ)|pβ′−1

]≤ N. (2.2.1)

(2) If H ≡ 0, then for all β′ ∈ (1, β), P-a.s. X·(·, ·), X−1· (·, ·) ∈ C([0,T ]2;Cβ

′

loc(Rd; Rd)) and

for all ϵ > 0 and p ≥ 2, there is a constant N = N(d, p,N0,T, β′, ϵ) such that

E[sups,t≤T|r−(1+ϵ)

1 Xt(s)|p0

]+ E

[sups,t≤T|r−ϵ1 ∇Xt(s)|pβ′−1

]≤ N

and


1 X−1t (s)|p0

]+ E

[sups,t≤T|r−ϵ1 ∇X−1

t (s)|pβ′−1

]≤ N.

Remark 2.2.2. The estimate (2.2.1) is used in Chapter 3 to take the optional projectionof a linear transformation of the inverse flow of a jump SDE driven by two independent


Weiner processes and two independent Poisson random measures relative to the filtrationgenerated by one of the Weiner processes and Poisson random measures.

Now, let us state our strong limit theorem for a sequence of flows, which will alsobe proved in the next section. We will use this strong limit theorem in [LM14a] to showthat the inverse flow of a jump SDE solves a parabolic SIDE. For each n, consider thestochastic flow X(n)

t = X(n)t (τ, x), (t, x) ∈ [0,T ] × Rd, generated by the SDE

dX(n)t = b(n)

t (X(n)t )dt + σ(n)lϱ

t (X(n)t )dwϱ

t +

∫Z

H(n)t (X(n)

t− , z)q(dt, dz), τ ≤ t ≤ T,

X(n)t = x, t ≤ τ.

Here we assume that for all n, b(n), σ(n), and H(n) satisfy the same measurability conditionsas b, σ, and H, respectively.

Theorem 2.2.3. Let Assumption 2.2.1(β) hold for some β > 1 and assume that b(n), σ(n),and H(n) satisfy Assumption 2.2.1 (β) uniformly in n ∈ N. Moreover, assume that

dPdt − limn→∞

(|r−1

1 b(n)t − r−1

1 bt|0 + |∇b(n)t − ∇bt|β−1

)= 0,

dPdt − limn→∞

(|r−1

1 σ(n)t − r−1

1 σt|β−1 + |∇σ(n)t − ∇σt|0

)= 0,

and for all (ω, t, z) ∈ Ω × [0,T ] × Z and n ∈ N,

|r−11 H(n)

t (z) − r−11 Ht(z)|0 + |∇H(n)

t (z) − ∇Ht(z)|β−1 ≤ K(n)(t, z),

where (K(n)t (z))n∈N is a sequence of R+-valued PT ⊗ Z measurable functions defined on

Ω × [0,T ] × Z satisfying for all (ω, t, z) ∈ Ω × [0,T ] × Z and n ∈ N,

K(n)t (z) +

∫Z

K(n)t (z)2π(dz) ≤ N0

anddPdt − lim

n→∞

∫Z

K(n)t (z)2π(dz) = 0.

Then for any stopping time τ ≤ T, β′ ∈ [1, β), and all ϵ > 0, and p ≥ 2, we have

limn→∞

(E

[supt≤T|r−(1+ϵ)

1 X(n)t (τ) − r−(1+ϵ)

1 Xt(τ)|p0

]+ E

[supt≤T|r−ϵ1 ∇X(n)

t (τ) − r−ϵ1 ∇Xt(τ)|pβ′−1

])= 0,

limn→∞


1 X(n);−1t (τ) − r−(1+ϵ)

1 X−1t (τ)|p0

]= 0,

and

limn→∞

E[supt≤T|r−ϵ1 ∇X(n);−1

t (τ) − r−ϵ1 ∇X−1t (τ)|pβ′−1

]= 0.


Let us introduce our class of solutions for the equation (2.1.1). LetOT be the F-optionalsigma-algebra on Ω× [0,T ]. For a each number β′ > 2, let Cβ

′

cts(Rd; Rd) be the linear spaceof all random fields v : Ω × [0,T ] × Rd → Rd such that v is OT ⊗ B(Rd)-measurable andP-a.s. r−λ1 (·)v·(·) is a C([0,T ];Cβ

′

(Rd; Rd)) for a real number λ > 0.We introduce the following assumption for a real number β > 2.

Assumption 2.2.2 (β). There is a constant N0 such that for all (ω, t) ∈ Ω × [0,T ],

|r−11 bt|0 + |r−1

1 σt|0 + |∇bt|β−1 + |∇σt|β−1 ≤ N0.

Theorem 2.2.4. Let Assumption 2.2.2(β) hold for some β > 2. Then for any stopping timeτ ≤ T and β′ ∈ [1, β), there exists a unique process u(τ) in Cβ

′

cts(Rd; Rd) that solves (2.1.2).Moreover, P-a.s. ut(τ, x) = Y−1

t (τ, x) for all (t, x) ∈ [0,T ]×Rd and for all ϵ > 0 and p ≥ 2,there is a constant N = N(d, p,N0,T, β′, ϵ) such that


1 ut(s)|p0

]+ E

[sups,t≤T|r−ϵ1 ∇ut(s)|pβ′−1

]≤ N.

Remark 2.2.5. It is clear by the proof of this theorem that if σ ≡ 0, then we only need toassume that Assumption 2.2.2 (β) holds for some β > 1.

Now let us consider the SPDE given by

dut(x) =(12σ

iϱt (x)σ jϱ

t (x)∂i jut(x) + bit(x)∂iut(x)

)dt + σiϱ

t (x)∂iut(x)dwϱt , τ < t ≤ T,

ut(x) = x, t ≤ τ. (2.2.2)

This SPDE differs from the one given in (2.1.2) by the first-order coefficient in thedrift. In order to obtain an existence and uniqueness theorem for this equation, we have toimpose additional assumptions on σ.

We introduce the following assumption for a real number β > 2.

Assumption 2.2.3 (β). There is a constant N0 > 0 such that for all (ω, t) ∈ Ω × [0,T ],

|r−11 bt|0 + |∇bt|β−1 + |σt|β+1 ≤ N0.

For each τ ≤ T , consider the stochastic flow Yt = Yt(τ, x), (t, x) ∈ [0,T ]×Rd, generatedby the SDE

dYt = −bt(Yt)dt − σϱt (Yt)dwϱ

t , τ < t ≤ T,

Yt = x, t ≤ τ.

2.3. Properties of stochastic flows 13

If Assumption 2.2.3(β) holds for some β > 2, then for all (ω, t, x) ∈ Ω × [0,T ] × Rd,

|bt(x)| ≤ |bt(x)| + |σt(x)|∇σt(x)| ≤ N0(N0 + 1) + N0|x|

and|∇bt|β−1 ≤ |∇bt|β−1 + |σt|β−1|∇

2σt|β−1 + |∇σt|2β−1 ≤ N0 + 2N2

0 ,

which immediately implies the following corollary of Theorem 2.2.4.

Corollary 2.2.6. If Assumption 2.2.3(β) holds for some β > 2, then for any stopping timeτ ≤ T and β′ ∈ [1, β), there exists a unique process u(τ) in Cβ

′

cts(Rd; Rd) that solves (2.2.2).Moreover, P-a.s. ut(τ, x) = Y−1

t (τ, x) for all (t, x) ∈ [0,T ]×Rd and for all ϵ > 0 and p ≥ 2,there is a constant N = N(d, p,N0,T, β′, ϵ) such that


1 ut(s)|p0

]+ E

[sups,t≤T|r−ϵ1 ∇ut(s)|pβ′−1

]≤ N.

2.3 Properties of stochastic flows

2.3.1 Homeomorphism property of flows

In this subsection, we collect some results about flows of jump SDEs that we will need. Inparticular, we present sufficient conditions that guarantee the homeomorphism property offlows of jump SDEs. First, let us introduce the following assumption, which is the usuallinear growth and Lipschitz condition on the coefficients b, σ, and H of the SDE (2.1.1).

Assumption 2.3.1. There is a constant N0 > 0 such that for all (ω, t, x, y) ∈ Ω×[0,T ]×R2d,

|bt(x)| + |σt(x)| ≤ N0(1 + |x|),

|bt(x) − bt(y)| + |σt(x) − σt(y)| ≤ N0|x − y|.

Moreover, for all (ω, t, x, y, z) ∈ Ω × [0,T ] × R2d × Z,

|Ht(x, z)| ≤ K1(t, z)(1 + |x|),

|Ht(x, z) − Ht(y, z)| ≤ K2(t, z)|x − y|,

where K1,K2 : Ω × [0,T ] × Z → R+ are PT ⊗Z-measurable functions satisfying

K1(t, z) + K2(t, z) +∫

Z

(K1(t, z)2 + K2(t, z)2

)π(dz) ≤ N0,

for all (ω, t, z) ∈ Ω × [0,T ] × Z.


It is well-known that under this assumption, there exists a unique strong solutionXt(s, x) of (2.1.1) (see e.g. Theorem 3.1 in [Kun04]). We will also make use of the follow-ing assumption.

Assumption 2.3.2. For all (ω, t, x, z) ∈ Ω × [0,T ] × Rd × Z, Ht(x, z) is differentiable in x,and there are constants η ∈ (0, 1) and Nκ > 0 such that for all (ω, t, x, z) ∈ (ω, t, x, z) ∈Ω × [0,T ] × Rd × Z : |∇Ht (ω, x, z)| > η,∣∣∣(Id + ∇Ht(x, z))−1

∣∣∣ ≤ Nκ.

The coming lemma shows that under Assumptions 2.3.1 and 2.3.2, the mapping x +Ht(x, z) from Rd to Rd is a diffeomorphism and the gradient of inverse map is bounded.

Lemma 2.3.1. Let Assumptions 2.3.1 and 2.3.2 hold. For all (ω, t, z) ∈ Ω× [0,T ]× Z, themapping Ht(·, z) : Rd → Rd defined by Ht(x, z) := x + Ht(x, z) is a diffeomorphism and

|H−1t (x, z)| ≤ NN0 + N |x| and |∇H−1

t (x, z)| ≤ N,

where N := (1 − η)−1 ∨ N0.

Proof. (1) On the set (ω, t, x, z) ∈ (ω, t, x, z) ∈ Ω × [0,T ] × Rd × Z : |∇Ht(ω, x, z)| ≤ η,we have

|κt(ω, x, z)| ≤

∣∣∣∣∣∣∣Id +

∞∑n=1

(−1)n[∇Ht(ω, x, z)]n

∣∣∣∣∣∣∣ ≤ 11 − η

.

It follows from Assumption 2.3.2 that for all ω, t, x, and z, the mapping ∇Ht(x, z) has abounded inverse. Therefore, by Theorem 0.2 in [DMGZ94], the mapping Ht(·, z) : Rd →

Rd is a global diffeomorphism. Moreover, for all ω, t, x and z,

|H−1t (x, z) − H−1

t (y, z)| ≤ N |x − y|,

which yields

|Ht(x, z) − Ht(y, z)| ≥ N−1|x − y| =⇒ |Ht(x, z)| + K1(t, z) ≥ N−1|x|,

and hence|H−1

t (x, z)| ≤ NK1(t, z) + N|x| ≤ NN0 + N |x|.

The following estimates are essential in the proof of the homeomorphic property ofthe flow and the derivation of moment estimates of the inverse flow. We refer the readerto Theorem 3.2 and Lemmas 3.7 and 3.9 in [Kun04] and Lemma 4.5.6 in [Kun97] (H ≡ 0case) for the proof of the following lemma.


Lemma 2.3.2. Let Assumption 2.3.1 hold.

(1) For all p ≥ 2, there is a constant N = N(p,N0,T ) such that for all s, s ∈ [0,T ] andx, y ∈ Rd,

E[supt≤T

r1(Xt(s, x))p

]≤ Nr1(x)p, (2.3.1)

E[supt≤T|Xt(s, x) − Xt(s, y)|p

]≤ N|x − y|p. (2.3.2)

(2) If Assumption 2.3.2 holds, then for any p ∈ R, there is a constant N = N(p,N0,T,η,Nκ) such that for all s ∈ [0,T ] and x, y ∈ Rd,

E[supt≤T

r1(Xt(s, x))p

]≤ Nr1(x)p, (2.3.3)

and

E[supt≤T|Xt(s, x) − Xt(s,Y)|p

]≤ N |x − y|p. (2.3.4)

In the next proposition, we collect some facts about the homeomorphic property ofthe flow. Let us mention that the homeomorphism property has been shown in [QZ08] tohold under a log-Lipschitz condition on the coefficients. The key idea is to use Bihari’sinequality instead of Gronwall’s inequality, but we do not pursue this here.

Proposition 2.3.3. Let Assumptions 2.3.1 and 2.3.2 hold.

(1) There exists a modification of the strong solution Xt(s, x), (s, t, x) ∈ [0,T ]2 × Rd,of (2.1.1), also denoted by Xt(s, x), that is càdlàg in s and t and continuous in x.Moreover, for any stopping time τ ≤ T, P-a.s. for all t ∈ [0,T ], the mappingsXt(τ, ·), Xt−(τ, ·) : Rd → Rd are homeomorphisms and the inverse of Xt(τ, ·), denotedby X−1

t (τ, ·), is càdlàg in t and continuous in x, and X−1t− (τ, ·) coincides with the inverse

of Xt−(τ, ·). In particular, if (xn)n≥1 is a sequence in Rd such that limn→∞ xn = x forsome x ∈ Rd, then P-a.s.

limn→∞

supt≤T|X−1

t (τ, xn) − X−1t (τ, x)| = 0.

Furthermore, for all β′ ∈ [0, 1), P-a.s. X(τ, ·) ∈ D([0,T ];Cβ′

loc(Rd; Rd)) and for all

ϵ > 0 and p ≥ 2, there is a constant N = N(d, p,N0,T, β′, ϵ) such that


1 Xt(τ)|pβ′

]≤ N. (2.3.5)

(2) If H ≡ 0, then P-a.s. for all s, t ∈ [0,T ], the Xt(s, x) and X−1t (s, x) are continuous in

s, t, and x. Moreover, for all β′ ∈ [0, 1], P-a.s. X(·, ·) ∈ C([0,T ]2;Cβ′

loc(Rd; Rd)) and for


all ϵ > 0 and p ≥ 2, there is a constant N = N(d, p,N0,T, β′, ϵ) such that


1 Xt(s)|pβ′

]≤ N. (2.3.6)

Proof. (1) Owing to Assumptions 2.3.1 and 2.3.2, by Lemma 2.3.1, for all ω, t and z, theprocess Ht(x, z) := x + Ht(x, z) is a homeomorphism (in fact, it is a diffeomorphism) in xand H−1

t (x, z) has linear growth and is Lipschitz. This implies that assumptions of Theorem3.5 in [Kun04] hold and hence there is modification of Xt(s, x), denoted Xt(s, x), such thatfor all s ∈ [0,T ], P-a.s. for all t ∈ [0,T ], Xt(s, ·) is a homeomorphism. Following [Kun04],for each (s, t, x) ∈ [0,T ]2 × Rd, we set

Xt(s, x) =

x t ≤ sXt(0, X−1

s (0, x)) t ≥ s,(2.3.7)

and remark that P-a.s. Xt(s, x) is càdlàg in s and t and continuous in x, and P-a.s. forall (s, t) ∈ [0,T ]2, Xt(s, ·) is a homeomorphism, and Xt(s, x) is a version of Xt(s, x) (theequation started at s). Fix a stopping time τ ≤ T . We will now show that Xt(τ, x) =Xt(s, x)|s=τ (i.e. Xt(s, x) evaluated at s = τ) is a version of Xt(τ, x). Define the sequence ofstopping times (τn)n≥1 by

τn =

n−1∑k=1

kTn

1(k−1)T

n ≤τ< kTn

+ T1τ≥ (n−1)T

n

.

For each n and x, let X(n)t = X(n)

t (x) = Xt(τn, x), t ∈ [0,T ]. It follows that for all n, t, and x,P-a.s. for all k ∈ 1, . . . , n,

X(n)t (x)1τn=

kTn = Xt

(kTn, x

)1τn=

kTn ,

and hence

X(n)t (x)1τn=

kTn = 1τn=

kTn

x + 1τn=kTn

∫] kT

n ,kTn ∨t]

br(X(n)r (x))dr

+ 1τn=kTn

∫] kT

n ,kTn ∨t]

σϱr (X(n)

r (x))dwϱr

+ 1τn=kTn

∫] kT

n ,kTn ∨t]

∫Z

Hr(X(n)r (x), z)q(dr, dz).

Since Ω is the disjoint union of the setsτn =

kTn

, k ∈ 1, . . . , n, it follows that X(n)

t (x)


solves

X(n)t (x) = x +

∫]τn,τn∨t]

br(X(n)r (x))dr +

∫]τn,τn∨t]

σϱr (X(n)

r (x))dwϱr

+

∫]τn,τn∨t]

∫Z

Hr(X(n)r (x), z)q(dr, dz).

Thus, by uniqueness, we find that for all t and x, P-a.s. Xt(τn, x) = X(n)t (x) = Xt(τn, x).

It is easy to check that for all t and x, P-a.s. Xt(τn, x) converges to Xt(τ, x) as n tends toinfinity. Since Xt(s, x) is càdlàg in s, Xt(τn, x) converges to Xt(τ, x) as n tends to infinity.Therefore, Xt(τ, x) is a version of Xt(τ, x) for all t and x. We identify Xt(s, x) and Xt(s, x)for all (s, t, x) ∈ [0,T ]2 × Rd. Using Lemma 2.3.2(1) and Corollary 2.5.3, we obtain thatP-a.s X·(τ, ·) ∈ D([0,T ];Cβ

′

loc(Rd; Rd)) and that the estimate (2.3.5) holds. Note here that

for all β ≥ 0, the Fréchet spaces D([0,T ];Cβloc(Rd; Rd)) and Cβloc(R

d; D([0,T ]; Rd)) areequivalent. It follows from the proof of Theorem 3.5 in [Kun04] that for every stoppingtime τ ≤ T , P-a.s.

lim|x|→∞

inft≤T|Xt(τ, x)| = ∞. (2.3.8)

Let (tn) ⊆ [0,T ] and (xn) ⊆ Rd be convergent sequences with limits t and x, respectively.First, assume tn < t for all n. By (2.3.8), for every stopping time τ ≤ T , P-a.s. the sequence(X−1

tn (τ, xn))

is uniformly bounded. Since P-a.s. X·(τ, ·) ∈ D([0,T ];Cβ(Rd; Rd)), β′ ∈ (0, 1),we have

limn→∞

(Xt−(τ, X−1

tn (τ, xn)) − Xt−(τ, X−1t− (τ, x)

)= lim

n→∞

(Xt−(τ, X−1

tn (τ, xn)) − x)

= limn→∞

(Xtn(τ, X

−1tn (τ, xn)) − x

)= lim

n→∞(xn − x) = 0,

which implieslimn→∞

X−1tn (τ, xn) = X−1

t− (τ, x).

A similar argument is used for tn > t. (2) It follows from the definition (2.3.7) that Xt(s, x)and X−1

t (s, x) are continuous in s, t, and x. Moreover, applying Lemma 2.3.2(1) and Corol-lary 2.5.3, we conclude that P-a.s. X·(·, ·) ∈ C([0,T ]2;Cβ

′

loc(Rd; Rd)) and that the estimate

(2.3.6) holds. The continuity of Xs(τ, x) with respect to s plays an important role in theproof of Theorem 2.2.4.

2.3.2 Moment estimates of inverse flows: Proof of Theorem 2.2.1

In this subsection, under Assumption 2.2.1(β), β ≥ 1, we derive moment estimates for theflow Xt(τ, x) and its inverse X−1

t (τ, x) in weighted Hölder norms and complete the proof ofTheorem 2.2.1. In particular, we will apply Corollaries 2.5.2 and 2.5.3 with the Banach


spaces V = D([0,T ]; Rd) and V = C([0,T ]2; Rd).

Proposition 2.3.4. Let Assumption 2.2.1(β) hold for some β > 1

(1) For all stopping time sτ ≤ T and β′ ∈ [1, β), P-a.s. ∇X·(τ, ·) ∈ D([0,T ];Cβ′−1

loc (Rd; Rd))and for all ϵ > 0 and p ≥ 2, there is a constant N = N(d, p,N0,T, β′, ϵ) such that

E[supt≤T|r−ϵ1 ∇Xt(τ)|pβ′−1

]≤ N. (2.3.9)

Moreover, for all p ≥ 2, there is a constant N = N(d, p,N0, β,T ) such that for allmulti-indices γ with 1 ≤ |γ| ≤

[β]

and all x ∈ Rd,

E[supt≤T|∂γXt(τ, x)|p

]≤ N (2.3.10)

and for all multi-indices γ with |γ| = [β]− and all x, y ∈ Rd,

E[supt≤T|∂γXt(τ, x) − ∂γXt(τ, y)|p

]≤ N |x − y|β

+p. (2.3.11)

(2) If H ≡ 0, then for all β′ ∈ [1, β), P-a.s. ∇X·(·, ·) ∈ C([0,T ]2;Cβ′−1

loc (Rd; Rd)) and for allϵ > 0 and p ≥ 2, there is a constant N = N(d, p,N0,T, β′, ϵ) such that


1 ∇Xt(s)|pβ′−1

]≤ N.

Moreover, for all p ≥ 2, there is a constant N = N(d, p,N0,T, β) such that for allmulti-indices γ with |γ| = [β]− and all s, s ∈ [0,T ] and x ∈ Rd,

E[supt≤T|∂γXt(s, x) − ∂γXt(s, x)|p

]≤ N |s − s|p/2. (2.3.12)

Proof. (1) Fix a stopping time τ ≤ T and write Xt(τ, x) = Xt(x). First, let us assumethat [β]− = 1. It follows from Theorem 3.4 in [Kun04] that P-a.s. for all t, Xt(τ, ·) iscontinuously differentiable and Ut = ∇Xt(τ, x) satisfies

dUt = ∇bt(Xt)Utdt + ∇σϱt (Xt−)Utdwϱ

t +

∫Z∇Ht(Xt−, z)Ut−q(dt, dz), τ < t ≤ T,

∇Xt = Id, t ≤ τ, (2.3.13)

where Id is the d × d-dimensional identity matrix. Taking λ = 0 in the estimates (3.10)and (3.11) in Theorem 3.3 in [Kun04], we obtain (2.3.10) and (2.3.11). Then applyingCorollary 2.5.3 with V = D([0,T ]; Rd), we have that X·(·) ∈ D([0,T ];Cβ

′

loc(Rd; Rd)) and


that (2.3.9) holds. The proof for [β]− > 1 follows by induction (see, e.g. the proof ofTheorem 6.4 in [Kun97]).

(2) The estimate (2.3.12) is given in Theorem 4.6.4 in [Kun97] in equation (19). Theremaining items of part (2) then follow in exactly the same way as part (1) with the onlyexception being that we apply Corollary 2.5.3 with V = C([0,T ]2; Rd).

Lemma 2.3.5. Let Assumption 2.2.1(β) hold for some β > 1.

(1) For any stopping time τ ≤ T and β′ ∈ [1, β), P-a.s. ∇X·(τ, ·)−1 ∈ D([0,T ];Cβ′−1

loc

(Rd; Rd)) and for all p ≥ 2, there is a constant N = N(d, p,N0,T, η,Nκ) such that forall x, y ∈ Rd

E[supt≤T|∇Xt(τ, x)−1|p

]≤ N (2.3.14)

and

E[supt≤T|∇Xt(τ, x)−1 − ∇Xt(τ, y)−1|p

]≤ N|x − y|((β−1)∧1)p. (2.3.15)

(2) If H ≡ 0, then for all β′ ∈ [1, β), P-a.s. ∇X·(·, ·)−1 ∈ C([0,T ]2;Cβ′−1

loc (Rd; Rd)) and forall p ≥ 2, there is a constant N = N(d, p,N0,T ) such that for all s, s ∈ [0,T ] andx ∈ Rd,

E[supt≤T|∇Xt(s, x)−1 − ∇Xt(s, x)−1|p

]≤ N |s − s|p/2.

Proof. (1) Let τ ≤ T be a fixed stopping time and write Xt(τ, x) = Xt(x). Using Itô’sformula (see also Lemma 3.12 in [Kun04]), we deduce that Ut = ∇Xt(x)−1 satisfies

dUt = Ut(∇σ

ϱt (Xt−)∇σ

ϱt (Xt−(τ)) − ∇bt(Xt)

)dt − Ut∇σ

ϱt (Xt)dwϱ

t

−

∫Z

Ut−∇Ht(Xt−, z)(Id + ∇Ht(Xt−, z))−1q(dt, dz)

+

∫Z

Ut∇Ht(Xt−, z)2(Id + ∇Ht(Xt−, z))−1π(dz)dt, τ < t ≤ T,

Ut = Id, t ≤ τ. (2.3.16)

Since matrix inversion is a smooth mapping, the coefficients of the linear equation (2.3.16)satisfy the same assumptions as the coefficients of the linear equation (2.3.13), and hencethe derivation of the estimates (2.3.14) and (2.3.15) proceed in the same way as the anal-ogous estimates for (2.3.13). To see that P-a.s. X·(·)−1 ∈ D([0,T ];Cβ

′−1loc (Rd; Rd)), we

only need to note that P-a.s. X·(·) ∈ D([0,T ];Cβ′−1

loc (Rd; Rd)) and that matrix inversion is asmooth mapping. Part (2) follows with the obvious changes.

As an immediate corollary, we obtain the diffeomorphism property of the flow Xt(τ, x)under Assumption 2.2.1(β), β > 1.


Corollary 2.3.6. Let Assumption 2.2.1(β) hold.

(1) For all stopping times τ ≤ T and β′ ∈ [1, β) the mapping Xt(τ, ·) : Rd → Rd is aCβ′

loc(Rd; Rd)-diffeomorphism, P-a.s. X·(τ, ·), X−1

· (τ, ·) ∈ D([0,T ];Cβ′

loc(Rd; Rd)) and for

all t ∈ [0,T ], X−1t− (τ) coincides with the inverse of Xt−(τ).

(2) If H ≡ 0, then for all β′ ∈ [1, β), P-a.s. X·(·, ·), X−1· (·, ·) ∈ C([0,T ]2,C

β′

loc(Rd; Rd)).

Proof. (1) Fix a stopping time τ ≤ T and write Xt(τ, x) = Xt(x). It follows from Propo-sitions 2.3.3 and 2.3.4 that P-a.s. for all t, the mappings Xt(·), Xt−(·) : Rd → Rd arehomeomorphisms and X·(·) ∈ D([0,T ];Cβ

′

loc(Rd; Rd)). Moreover, by Lemma 2.3.5, P-a.s.

for all t and x, the matrix ∇Xt (τ, x) has an inverse. Therefore, by Hadamard’s Theorem(see, e.g., Theorem 0.2 in [DMGZ94]), P-a.s. for all t, Xt(·) is a diffeomorphism. Usingthe chain rule, P-a.s. for all t and x,

∇X−1t (x) = ∇Xt(X−1

t (x))−1. (2.3.17)

Since, by Lemma 2.3.5, P-a.s. [∇X·(·)]−1 ∈ D([0,T ];Cβ′−1

loc (Rd; Rd)) and we know thatP-a.s. for all t, X−1

t (·) is differentiable, it follows from (2.3.17) that P-a.s.

∇X·(X−1· (·))−1 ∈ D([0,T ];C(β′−1)∧1

loc (Rd; Rd)).

We proceed inductively to complete the proof. Making the obvious changes in the proofof part (1), we obtain part (2).

We conclude with a derivation of Hölder moment estimates of the inverse flow X−1t (τ, x),

which will complete the proof of Theorem 2.2.1.

Proof of Theorem 2.2.1. (1) Fix a stopping time τ ≤ T and write Xt(τ, x) = Xt(x). Fixϵ > 0. First, let us assume that

[β]−= 1. Set Jt(x) = | det∇Xt(x)|. It is clear from (2.3.10)

that for all p ≥ 2 and x, there is a constant N = N(d, p,N0,T ) such that

E[supt≤T|Jt(x)|p] ≤ N. (2.3.18)

By the mean value theorem, for all x and y and p ∈ R, we have

|r1(x) p − r1(y) p| ≤ |p|(r1(x) p−1 + r1(y) p−1)|x − y|. (2.3.19)

Using the change of variable (x, y) = (X−1t (x), X−1

t (y)), Fatou’s lemma, Fubini’s theorem,Hölder’s inequality, and the inequalities (2.3.3), (2.3.18), (2.3.19), (2.3.2), and (2.3.4), forall δ ∈ (0, 1] and p > d

ϵ, we conclude that there is a constant N = N(d, p,N0,T, δ, η,Nκ, ϵ)


such that

E supt≤T

∫Rd|r1(x)−(1+ϵ)X−1

t (x)|pdx ≤∫

Rd|x|pE sup

t≤T[r1(Xt(x))−p(1+ϵ)Jt(x)]dx

≤ NE∫

Rdr1(x)−pϵdx ≤ N

and

E supt≤T

∫|x−y|<1

|r−(1+ϵ)1 (x)X−1

t (x) − r−(1+ϵ)1 (y)X−1

t (y)|p

|x − y|2d+δp dxdy

≤

∫|x−y|<1

E supt≤T

r−p(1+ϵ)1 (Xt(x))|x − y|pJt(x)Jt(y)|Xt(x) − Xt(y)|2d+δp

dxdy

+

∫|x−y|<1

E supt≤T

|y|p|r−(1+ϵ)1 (Xt(x)) − r−(1+ϵ)

1 (Xt(y))|pJt(x)Jt(y)|Xt(x) − Xt(y)|2d+δp

dxdy

≤ N∫|x−y|<1

r1(x)−p(1+ϵ)

|x − y|2d−(1−δ)p dxdy + N∫|x−y|<1

r1(x)−p(1+ϵ) + r1(y)−p(1+ϵ)

|x − y|2d−(1−δ)p dxdy ≤ N.

Similarly, making use of the inequalities (2.3.3), (2.3.18), (2.3.19), (2.3.2), (2.3.4), (2.3.14),and (2.3.15), for all p > d

ϵ∨ d

β−β′∨ d

2−β′ , we get

E supt≤T

∫Rd|r−ϵ(x)∇X−1

t (x)|pdx ≤∫

RdE sup

t≤T[r1(Xt(x))−pϵ |[∇Xt(x)]−1|pJt(x)]dx

≤ NE∫

Rdr1(x)−pϵdx ≤ N

and

E supt≤T

∫|x−y|<1

|r−ϵ1 (x)∇X−1t (x) − r−ϵ1 (y)∇X−1

t (y)|p

|x − y|2d+(β′−1)p dxdy

≤

∫|x−y|<1

E supt≤T

[|r−ϵ1 (Xt(x))[∇Xt(x)]−1 − r−ϵ1 (Xt(y))[∇Xt(y)]−1|pJt(x)Jt(y)

|Xt(x) − Xt(y)|2d+(β′−1)p

]dxdy

≤ N∫|x−y|<1

r1(x)−pϵ

|x − y|2d−(β−β′)p dxdy + N∫|x−y|<1

r1(x)−pϵ + r1(y)−pϵ

|x − y|2d−(2−β′)p dxdy ≤ N,

where N = N(d, p,N0,T, β′, η,Nκ, ϵ) is a positive constant. Therefore, combining theabove estimates and applying Corollary 2.5.2, we find that for all p ≥ 2, there is f aconstant N = N(d, p,N0,T, β′, η,Nκ, ϵ), such that


1 X−1t (τ)|p0

]+ E


t (τ)|pβ′−1

]≤ N.

It is well-known that the the inverse map I on the set of invertible d × d-dimensional


matrices is infinitely differentiable and for all n, there is a constant N = N(n, d) suchthat for all invertible matrices M, the nth derivative of I evaluated at M, denoted I(n)(M),satisfies ∣∣∣I(n)(M)

∣∣∣ ≤ N |M−n−1| ≤ N∣∣∣M−1

∣∣∣n+1.

We claim that for all n and every multi-index γ with |γ| = n, the components of ∂γX−1t (x)

are a polynomial in terms of the entries of [∇Xt(X−1t (x))]−1 and ∂γ

′

∇Xt(X−1t (x)) for all

multi-indices γ′ with 1 ≤ |γ′| ≤ n − 1. Assume that statement holds for some n. By thechain rule, for all ω, t, and x, we have

∇(∇Xt(X−1t (x))−1) = I(1)(∇Xt(X−1

t (x)))∇2Xt(X−1t (x))∇Xt(X−1

t (x))−1

and for all multi-indices γ with 1 ≤ |γ′| ≤ n − 1, we have

∇(∂γ′

∇Xt(X−1t (x))) = ∂γ

′

∇2Xt(X−1t (x))∇Xt(X−1

t (x))−1,

where ∇2Xt(X−1t (x)) is the tensor of second-order derivatives of Xt(·) evaluated at X−1

t (x).This implies that for every multi-index γ with |γ| = n + 1, the components of ∂γX−1

t (x)are a polynomial in terms of the entries of ∇Xt(X−1

t (x))−1 and ∂γ′

∇Xt(X−1t (x)) for all multi-

indices γ′ with 1 ≤ |γ′| ≤ n. By induction, the claim is true. Therefore, for [β]− ≥ 2, using(2.3.10) and (2.3.11), we obtain the moment estimates for the inverse flow in the almostexact same way we did for [β]− = 1. Making the obvious changes in the proof of part (1),we obtain part (2). This completes the proof of Theorem 2.2.1.

2.3.3 Strong limit of a sequence of flows: Proof of Theorem 2.2.3

Proof of Theorem 2.2.3. Let τ ≤ T be a fixed stopping time and write Xt(x) = Xt(τ, x). Foreach n, let

Z(n)t (x) = X(n)

t (x) − Xt(x), (t, x) ∈ [0,T ] × Rd.

Throughout the proof we denote by (δn)n≥1 a deterministic sequence with δn → 0 as n→ ∞that may change from line to line. Let N = N(p,N0,T ) be a positive constant, which maychange from line to line. By virtue of Theorem 2.1 in [Kun04] and (2.3.1), for all p ≥ 2and t, x and n, we have

E[sups≤t|Z(n)

s (x)|p]≤ NE

∫]0,t]|Z(n)

s (x)|pds + Nδnr1(x)p.


limn→∞

E[supt≤T|r−ϵ1 ∇X(n);−1

t (τ) − r−ϵ1 ∇X−1t (τ)|pβ′−1

]= 0,

it suffices to show that for all x,

dP − limn→∞

supt≤T|X(n);−1

t (x) − X−1t (x)| = 0 (2.3.22)

anddP − lim

n→∞supt≤T|∇X(n);−1

t (x) − ∇X−1t (x)| = 0. (2.3.23)

For each n, define

Θ(n)t (x) = r1(X(n)

t (x))−1 − r1(Xt(x))−1, (t, x) ∈ [0,T ] × Rd.

For all ω, t, x, and n, we have

|Θ(n)t (x)| ≤ r1(X(n)

t (x))−1r1(Xt(x))−1|Z(n)t (x)|,

and hence using Hölder’s inequality, (2.3.4), and (2.3.20), we obtain that for all p ≥ 2, x,there is a constant N = N(p,N0,T, η,Nκ) such that for all n,

E[supt≤T|Θ

(n)t (x)|p

]≤ Nr1(x)−pδn,

where N = N(p,N0,T, η,Nκ) is a constant. Furthermore, since

|∇Θ(n)t (x)| ≤ r1(X(n)

t (x))−2|∇X(n)t (x)| + r1(Xt(x))−2|∇X(n)

t (x)|,

for all ω, t, x, and n, applying (2.3.4) and (2.3.10), for all p ≥ 2, x, and n, we get

E[supt≤T|r1(x)Θ(n)

t (x) − r1(y)Θ(n)t (y)|p

]≤ N|x − y|p.

Then owing to Corollary 2.5.4, for all p ≥ 2,

limn→∞

E[supt≤T|Θ

(n)t |

p0

]= 0. (2.3.24)

We claim that for all R > 0,

dP − limn→∞

E(n,R) := dP − limn→∞

supt≤T|X(n);−1

t − X−1t |0;|x|≤R = 0. (2.3.25)

Fix R > 0. It is enough to show that every subsequence of E(n) = E(n,R) has a sub-subsequence converging to 0, P-a.s.. Owing to (2.3.21) and (2.3.24), for a given subse-


quence (E(nk)), we can always find sub-subsequence (still denoted (E(nk)) to avoid doubleindices) such that P-a.s.,

limk→∞

supt≤T|X(nk)

t − Xt|β′;|x|≤R = 0, ∀ R > 0, (2.3.26)

andlimk→∞

supt≤T|r1(X(nk)

t (x))−1 − r1(Xt(x))−1|0 = 0.

Fix an ω for which both limits are zero. We will prove that

limk→∞

supt≤T|X(nk);−1

t (ω) − X−1t (ω)|0;|x|≤R = 0. (2.3.27)

Suppose, by contradiction, that (2.3.27) is not true. Then there exists an ε > 0 and asubsequence of (nk) (still denoted (nk)) such that tnk → t− (or tnk → t+) and xnk → x ask → ∞ with

∣∣∣xnk

∣∣∣ ≤ R such that (dropping ω),

|X(nk);−1tnk

(xnk) − X−1tnk

(xnk)| ≥ ε. (2.3.28)

Arguing by contradiction and using (2.3.3), we have

supk∈N|X(nk);−1

tnk(xnk)| < ∞. (2.3.29)

Applying (2.3.29), (2.3.26), and the fact that X·(·), X−1· (·) ∈ D([0,T ];Cβ

′

loc(Rd; Rd)) , we

obtain

limk→∞

(Xt−(X

(nk);−1tnk

(xnk)) − Xt−(X−1tnk

(xnk)))= lim

k→∞

(Xt−(X

(nk);−1tnk

(xnk)) − xnk

)= lim

k→∞

(Xt−(X

(nk);−1tnk

(xnk)) − X(nk)tnk

(X(nk);−1tnk

(xnk)))

= limk→∞

(Xt−(X

(nk);−1tnk

(xnk)) − Xtnk(X(nk);−1

tnk(xnk))

)+ lim

k→∞

(Xtnk

(X(nk);−1tnk

(xnk)) − X(nk)tnk

(X(nk);−1tnk

(xnk)))= 0,

which contradicts (2.3.28), and hence proves (2.3.27), (2.3.25), and (2.3.22). For each n,define

U (n)t = U (n)(t, x) = ∇X(n)

t (x)−1 and U(t) = U(t, x) = ∇Xt(x)−1, (t, x) ∈ [0,T ] × Rd.

2.4. Classical solutions of degenerate SPDEs: Proof of Theorem 2.2.4 26

Using (2.3.14) and (2.3.15) and repeating the arguments given above, for all p ≥ 2, we get

limn→∞

E[supt≤T|r−ϵ1 U (n)

t − r−ϵ1 Ut|pβ′−1

]= 0.

Then (2.3.3) and (2.3.25) imply that for all R > 0,

dP − limn→∞

supt≤T|∇X(n);−1

t (x) − ∇X−1t (x)|0;|x|≤R

= dP − limn→∞

supt≤T|∇X(n)

t (X(n);−1t (x))−1 − ∇Xt(X−1

t (x))−1|0;|x|≤R = 0,

which yields (2.3.23) and completes the proof.

2.4 Classical solutions of degenerate SPDEs: Proof of The-orem 2.2.4

Proof of Theorem 2.2.4. Fix a stopping time τ ≤ T . By virtue of Theorem 2.2.1, we onlyneed to show that Y−1(τ) = Y−1

t (τ, x) solves (2.1.2) and is the unique solution. Suppose wehave shown that Y−1(s, x), s ∈ [0,T ], solves (2.1.2) (i.e. where τ = s is deterministic). It isthen a routine argument to conclude that Y−1(τ′) solves (2.1.2) for finite-valued stoppingtimes τ′ ≤ T . We can then use an approximation argument as in the proof of Proposition2.3.3 to show that Y−1(τ) = Y−1

t (τ, x) solves (2.1.2). Thus, it suffices to take τ deterministic.Let ut(x) = ut(s, x) = Y−1

t (s, x), (s, t, x) ∈ [0,T ]2 × Rd. Fix (s, t, x) ∈ [0,T ]2 × Rd withs < t and write Yt(x) = Yt(s, x). Let ((tM

n )0≤n≤M)1≤M≤∞ be a sequence of partitions of theinterval [s, t] such that for all M > 0, (tM

n )0≤n≤M has mesh size (t − s)/M. Fix M and set(tn)0≤n≤M = (tM

n )0≤n≤M. Immediately, we obtain

ut(x) − x =M−1∑n=0

(utn+1(x) − utn(x)). (2.4.1)

We will use Taylor’s theorem to expand each term in the sum on the right-hand-side of(2.4.1). By Taylor’s theorem, for all n and y, we have

utn+1(Ytn+1(y)) − utn(Ytn+1(y)) = y − utn(Ytn+1(y)) = utn(Ytn(y)) − utn(Ytn+1(y))

= ∇utn(Ytn(y))(Ytn(y) − Ytn+1(y)) − (Ytn(y) − Ytn+1(y))∗Θn(Ytn(y))(Ytn(y) − Ytn+1(y)),

(2.4.2)

where

Θi jn (z) =

∫ 1

0(1 − θ)∂i jutn

(z + θ(Ytn+1(Y

−1tn (z)) − z)

)dθ.


Since for all n, Ytn+1(s, x) = Ytn+1(tn,Ytn(s, x)), we have

Ytn+1(Y−1tn (x)) = Ytn+1(tn, x)

and hence substituting y = Y−1tn (x) into (2.4.2), for all n, we get

utn+1(x) − utn(x) = An + Bn, (2.4.3)

where

An := ∇utn(x)(x − Ytn+1(tn, x)) − (x − Ytn+1(tn, x))∗Θi jn (x)(x − Ytn+1(tn, x))

andBn := (utn+1(x) − utn(x)) − (utn+1(Ytn+1(tn, x)) − utn(Ytn+1(tn, x))).

Applying Taylor’s theorem once more, for all n, we obtain

Bn = Cn + Dn, (2.4.4)

whereCn := (∇utn+1(x) − ∇utn(x))(x − Ytn+1(tn, x)),

Dn := −(x − Ytn+1(tn, x))∗Θn(x)(x − Yti+1(ti, x))),

and

Θn(x)i j :=∫ 1

0(1 − θ)∂i j(utn+1 − utn)(x + θ(Ytn+1(tn, x) − x))dθ.

Thus, combining (2.4.1), (2.4.3), and (2.4.4), P-a.s. we have

ut(x) − x =M−1∑n=0

(An +Cn + Dn). (2.4.5)

Now, we will derive the limit of the right-hand-side of (2.4.5).

Claim 2.4.1. (1)

dP − limM→∞

M−1∑n=0

An = −

∫]s,t]

[12σiϱ

r (x)σ jϱr (x)∂i jur(x) + bi

r(x)∂iur(x)]dr

−

∫]s,t]

σiϱr (x)∂iur(x)dwϱ

r ;

(2) dP − limM→∞∑M−1

n=0 Dn = 0;(3) dP − limM→∞

∑M−1n=0 Cn =

∫]s,t]

σjϱr (x)∂ jσ

iϱr (x)∂iur(x)dr +

∫]s,t]

σiϱr (x)σ jϱ

r (x)∂i jur(x)dr.


Proof of Claim 2.4.1. (1) For all n, we have

∇utn(x)(x − Ytn+1(tn, x)

)= −

∫]tn,tn+1]

bir(x)∂iutn(x)dr −

∫]tn,tn+1]

σiϱr (x)∂iutn(x)dwϱ

r

+ R(1)n + R(2)

n ,

whereR(1)

n :=∫

]tn,tn+1]

(bi

r(x) − bir(Yr(tn, x))

)∂iutn(x)dr

andR(2)

n :=∫

]tn,tn+1][σiϱ

r (x) − σiϱr (Yr(tn, x))]∂iutn(x)dwϱ

r .

Since b and σ are Lipschitz, there is a constant N = N(N0,T ) such that

M−1∑n=0

∣∣∣R1n

∣∣∣ ≤ N sups≤r≤t|∇ur(x)| sup

|r1−r2 |≤t

M

|x − Yr1(r2, x)|

and ∫]s,t]

∣∣∣∣∣∣∣M−1∑n=0

1]tn,tn+1](r)(σi·

r (x) − σi·r (Yr(tn, x))

)∂iutn(x)

∣∣∣∣∣∣∣2

ds

≤ N sups≤r≤t|∇ur(x)|2 sup

|r1−r2 |≤t

M

|x − Yr1(r2, x)|2.

Owing to the joint continuity of Yt(s, x) in s and t and the dominated convergence theoremfor stochastic integrals, we obtain

dP − limM→∞

M−1∑n=0

(R(1)n + R(2)

n ) = 0. (2.4.6)

In a similar way, this time using the continuity of ∇ut(x) in t and the linear growth of band σ, we find

dP − limM→∞

M−1∑n=0

(−

∫]tn,tn+1]

bir(x)∂iutn(x)dr −

∫]tn,tn+1]

σiϱr (x)∂iutn(x)dwϱ

r

)

= −

∫]s,t]

br(x)∂iur(x)dr −∫

]s,t]σϱ

r (x)∂iur(x)dwϱr .

For all n, we have

−(x − Ytn+1(tn, x))∗Θn(x)(x − Ytn+1(tn, x)) = S (1)n + S (2)

n ,


where S (1)n (t, x) has only drdr and drdwϱ

r terms and where

S (2)n : = −

12

(∫]tn,tn+1]

σiϱr (Yr(tn, x))dwϱ

r

)∂i jutn(x)

(∫]tn,tn+1]

σ jϱr (Yr(tn, x))dwϱ

r

)−

(∫]tn,tn+1]

σiϱr (Yr(tn, x))dwϱ

r

) (Θi j

n (x) −12∂i jutn(x)

) (∫]tn,tn+1]

σ jϱr (Yr(tn, x))dwϱ

r

).

Since ∣∣∣∣∣Θi jn (x) −

12∂i jutn(x)

∣∣∣∣∣ =∣∣∣∣∣∣∫ 1

0(1 − θ)(∂i jutn(x + θ(Ytn+1(tn, x) − x)) − ∂i jutn(x))dθ

∣∣∣∣∣∣≤ N sup

|r1−r2 |≤t

M ,θ∈(0,1)|∂i jur1(x + θ(Yr2(r1, x) − x)) − ∂i jur1(x))|,

proceeding as in the derivation of (2.4.6) and using the joint continuity of ∂i jut(x) in t andx, the continuity of Yt(s, x) in s and t, the explicit form of the quadratic variation of thestochastic integral (i.e. part (5) of Chapter 2, Section 2, Theorem 2 in [LS89]), and thestochastic dominated convergence theorem, we obtain

dP − limM→∞

M−1∑n=0

S (2)n = −

12

∫]0,t]

σiϱr (x)σ jϱ

r (x)∂i jur(x)dr.

Similarly, making use of the properties stated in Theorem 2.2.1(2), we have

dP − limM→∞

M−1∑n=0

S (1)n = 0,

which completes the proof of part (1). The proof of part (2) is similar to the proof of part(1), so we proceed to the proof of part (3). We know that for all n, Ytn+1(x) = Ytn+1(tn,Ytn(x)).Thus, for all n, we have utn+1(x) = utn(Y

−1tn+1

(tn, x)), and hence by the chain rule,

∇utn+1(x) = ∇utn(Y−1tn+1

(tn, x))∇Y−1tn+1

(tn, x). (2.4.7)

By (2.4.7) and Taylor’s theorem, for all n, we have

Cn = (∇utn+1(x) − ∇utn(x))(x − Ytn+1(tn, x))

= ∇utn(Y−1tn+1

(tn, x))(∇Y−1tn+1

(tn, x) − Id)(x − Ytn+1(tn, x))

+(Y−1tn+1

(tn, x) − x)∗Θn(x)(x − Ytn+1(tn, x)) =: En + Fn,

where

Θi jn (x) :=

∫ 1

0∂i jutn(x + θ(Y−1

tn+1(tn, x) − x))dθ.


Using Itô’s formula, for all n, we get (see, also, Lemma 3.12 in [Kun04]),

∇Ytn+1(tn, x)−1 = Id −

∫]tn,tn+1]

∇Yr(tn, x)−1∇σϱr (Yr(tn, x))dwϱ

r

+

∫]tn,tn+1]

∇Yr(tn, x)−1 (∇σϱ

r (Yr(tn, y))∇σϱr (Yr(tn, x)) − ∇br(Yr(tn, x))

)dr,

and hence

∇Y−1tn+1

(tn) − Id = ∇Y−1tn+1

(tn,Y−1tn+1

(tn, x)) − Id =: G(1)tn,tn+1

(Y−1tn+1

(tn, x)) +G(2)tn,tn+1

(Y−1tn+1

(tn, x)),

where for y ∈ Rd,

G(1)tn,tn+1

(y) :=∫

]tn,tn+1]∇Yr(tn, z)−1 (

∇σϱr (Yr(tn, y))∇σϱ

r (Yr(tn, y)) − ∇br(Yz(tn, y)))

dr

andG(2)

tn,tn+1(z) := −

∫]tn,tn+1]

∇Yr(tn, y)−1∇σϱr (Yr(tn, y))dwϱ

r .

By the Burkholder-Davis-Gundy inequality, Hölder’s inequality, and the inequalities (2.3.2),(2.3.14), and (2.3.15), for all p ≥ 2, there is a constant N = N(p, d,N0,T ) such that for allx1 and x2,

E[|G(2)

tn,tn+1(x1)|p

]≤ NM−p/2+1

∫]tn,tn+1]

E[|∇Yr(tn, x1)−1|p|∇σr(Yr(tn, x1)|p

]dr ≤ NM−p/2

and

E[|G(2)

tn,tn+1(x1) −G(2)

tn,tn+1(x2)|p

]≤ NM−p/2+1

∫]tn,tn+1]

E[|∇Yr(tn, x1)−1 − ∇Yr(tn, x2)−1|p

]dr

+NM−p/2+1∫

]tn,tn+1]

(E

[|∇Yr(tn, x1)−1|2p

])1/2 (E

[|Yr(tn, x1) − Yr(tn, x2)|2p

])1/2dr

≤ NM−p/2|x − y|p.

Thus, by Corollary 2.5.3, we obtain that for all p ≥ 2, ϵ > 0, and δ < 1, there is a constantN = N(p, d, δ,N0,T ) such that

E[|r−ϵG(2)

tn,tn+1|pδ

]≤ NM−p/2. (2.4.8)

For all n, we have

En = ∇utn(Y−1tn+1

(tn, x))G(1)tn,tn+1

(Y−1tn+1

(tn, x))(x − Ytn+1(tn, x))

+ ∇utn(x)G(2)tn,tn+1

(x)(x − Ytn+1(tn, x))


+ ∇utn(Y−1tn+1

(tn, x))(G(2)tn,tn+1

(Y−1tn+1

(tn, x)) −G(2)tn,tn+1

(x))(x − Ytn+1(tn, x))

+ (∇utn(Y−1tn+1

(tn, x) − ∇utn(x))G(2)tn,tn+1

(x)(x − Ytn+1(tn, x))

One can easily check that

dP − limM→∞

M−1∑n=0

∇utn(Y−1tn+1

(tn, x))G(1)tn,tn+1

(Y−1tn+1

(tn, x))(x − Ytn+1(tn, x)) = 0. (2.4.9)

Since ∇ut(x) is jointly continuous in t and x and Y−1t (s, x) is jointly in s and t, we have

dP − limM→∞

supn|∇utM

n(Y−1

tMn+1

(tMn , x)) − ∇utM

n(x)| = 0.

Moreover, using Hölder’s inequality, (2.4.8), and (2.3.1), we get

supM

EM−1∑n=0

|G(2)tn,tn+1

(x)|x − Ytn+1(tn, x)| < ∞,

and hence

dP − limM→∞

M−1∑n=0

(∇utn(Y−1tn+1

(tn, x)) − ∇utn(x))G(2)tn,tn+1

(x)(x − Ytn+1(tn, x)) = 0. (2.4.10)

We claim that

dP − limM→∞

M−1∑n=0

∇utn(Y−1tn+1

(tn, x))(G(2)

tn,tn+1(Y−1

tn+1(tn, x)) −G(2)

tn,tn+1(x)

)(x − Ytn+1(tn, x)) = 0.

(2.4.11)Set

JM =

M−1∑n=0

|G(2)tn,tn+1

(Y−1tn+1

(tn, x)) −G(2)tn,tn+1

(x)|x − Ytn+1(tn, x)|.

For all δ, ϵ ∈ (0, 1), we have

P(JM > δ) ≤ P(JM > δ, max

n|Y−1

tn+1(tn, x) − x| ≤ ϵ

)+ P

(max

n|Y−1

tn+1(tn, x) − x| > ϵ

).

By virtue of (2.4.8), there is a deterministic constant N = N(x) independent of M suchthat for all ω ∈ V M := maxn |Y−1

tn+1(tn, x) − x| ≤ ϵ,

JM ≤ NϵδM−1∑n=0

[r−ϵ1 G(2)tn,tn+1

]δ|x − Ytn+1(tn, x)|,


which implies that

E1V M JM ≤ NϵδEM−1∑n=0

([r−ϵ1 G(2)

tn,tn+1]2δ + |x − Ytn+1(tn, x)|2

)≤ Nϵδ

M−1∑n=0

M−1 ≤ Nϵδ.

Applying Markov’s inequality, we derive

P(JM | > δ, maxn|Y−1

tn+1(tn, x) − x| ≤ ϵ) ≤ N

ϵδ

δ,

and hence for all δ > 0,lim

M→∞P(JM > δ) = 0,

which yields (2.4.11). Owing to (2.4.9), (2.4.10), and (2.4.11) we have

dP − limM→∞

M−1∑n=0

En = limM→∞

M−1∑n=0

∇utn(x)G(2)tn,tn+1

(x)(x − Ytn+1(tn, x)).

Proceeding as in the proof of part (1) of the claim, we obtain

limM→∞

M−1∑n=0

Kn

= limM→∞

M−1∑n=0

∇utn(x)∫

]tn,tn+1](∇Yr(tn, x)−1 − Id)∇σϱ

r (x)dWϱr

∫]tn,tn+1]

σϱr (x)dWϱ

r

+ limM→∞

M−1∑n=0

∇utn(x)∫

]tn,tn+1]∇σϱ

r (x)dwϱr

∫]tn,tn+1]

σϱr (x)dwϱ

r

=

∫]s,t]

σ jϱr (x)∂ jσ

iϱr (x)∂iur(x)dr. (2.4.12)

It is easy to check that for all n,

Fn = (Y−1tn+1

(tn, x) − x)∗Θn(x)(x − Ytn+1(tn, x))

=: (G(3)tn,tn+1

(Y−1tn+1

(tn, x)) +G(4)tn,tn+1

(Y−1tn+1

(tn, x)))∗Θn(x)(x − Ytn+1(tn, x)),

where for y ∈ Rd,

G(3)tn,tn+1

(y) := −∫

]tn,tn+1]br(Yr(tn, y))dr, G(4)

tn,tn+1(y) := −

∫]tn,tn+1]

σϱr (Yr(tn, y))dwϱ

r .

2.5. Appendix 33

Arguing as in the proof of (2.4.12), we get

dP − limM→∞

M−1∑n=0

Fn =

∫]s,t]

σiϱr (x)σ jϱ

r (x)∂i jur(x)dt,

which completes the proof of the claim.

By virtue of (2.4.5) and Claim 2.4.1, for all s and t with s ≤ t and x, P-a.s.

ut(x) = x +∫

]s,t]

(12σiϱ

r (x)σ jϱ(x)∂i jur(x) − bit(x)∂iur(x)

)dr −

∫]s,t]

σiϱr (x)∂iur(x)dwϱ

r .

(2.4.13)Owing to Theorem 2.2.1, u = ut(x) has a modification that is jointly continuous in s and tand twice continuously differentiable in x. It is easy to check that the Lebesgue integral onthe right-hand-side of (2.4.13) has a modification that is continuous in s, t, and x. Thus, thestochastic integral on the right-hand-side of (2.4.13) has a modification that is continuousin s, t, and x, and hence the equality in (2.4.13) holds P-a.s. for all s and t with s ≤ t and x.This proves that Y−1(τ) = Y−1

t (τ, x) solves (2.1.2). However, if u1(τ), u2(τ) ∈ Cβ′

cts(Rd; Rd)are solutions of (2.1.2), then applying the Itô-Wentzell formula (see, e.g. Theorem 9 inChapter 1, Section 4.8 in [Roz90]), we get that P-a.s. for all t and x,

u1t (τ,Yt(τ, x)) = x = u2

t (τ,Yt(τ, x)),

which implies that P-a.s. for all t and x, u1(τ) = Y−1t (τ, x) = u2(τ). Thus, Y−1(τ) = Y−1

t (τ, x)is the unique solution of (2.1.2) in Cβ

′

cts(Rd; Rd).

2.5 Appendix

Let V be an arbitrary Banach space. The following lemma and its corollaries are indis-pensable in this chapter.

Lemma 2.5.1. Let Q ⊆ Rd be an open bounded cube, p ≥ 1, δ ∈ (0, 1], and f be aV-valued integrable function on Q such that

[f]δ;p;Q;V :=

(∫Q

∫Q

| f (x) − f (y)|pV|x − y|2d+δp dxdy

)1/p

< ∞.

Then f has a Cδ(Q; V)-modification and there is a constant N = N(d, δ, p) independent off and Q such that

[ f ]δ;Q;V ≤ N[f]δ,p;Q;V

2.5. Appendix 34

and

supx∈Q| f (x)|V ≤ N|Q|δ/d[ f ]δ;p;Q;V + |Q|−1/p

(∫Q| f (x)|pVdx

)1/p

,

where |Q| is the volume of the cube.

Proof. If V = R, then the existence of a continuous modification of f and the estimate of[f]δ;Q follows from Lemma 2 and Exercise 5 in Chapter 10, Section 1, in [Kry08]. The

proof for a general Banach space is the same. For all x ∈ Q, we have

| f (x)|V ≤1|Q|

∫Q| f (x) − f (y)|Vdy +

1|Q|

∫Q| f (y)|Vdy

≤ N1|Q|

[f]δ,p;Q

∫Q|x − y|δdy +

1|Q|

∫Q| f (y)|Vdy

≤ N |Q|δ/d[ f ]δ,p;Q + |Q|−1/p

(∫Q| f (y)|pVdy

)1/p

,

which proves the second estimate.

The following is a direct consequence of Lemma 2.5.1.

Corollary 2.5.2. Let p ≥ 1, δ ∈ (0, 1], and f be a V-valued function on Rd such that

| f |δ;p;V :=(∫

Rd| f (x)|pVdx +

∫|x−y|<1

| f (x) − f (y)|pV|x − y|2d+δp dxdy

)1/p

< ∞.

Then f has a Cδ(Rd; V)-modification and there is a constant N = N(d, δ, p) independentof f such that

| f |δ;V ≤ N | f |δ;p;V .

Corollary 2.5.3. Let X be a V-valued random field defined on Rd. Assume that for somep ≥ 1, l ≥ 0, and β ∈ (0, 1] with βp > d there is a constant N > 0 such that for allx, y ∈ Rd,

E[|X(x)|pV

]≤ Nr1(x)lp (2.5.1)

andE

[|X(x) − X(y)|pV

]≤ N[r1(x)lp + r1(y)lp]|x − y|βp. (2.5.2)

Then for all δ ∈ (0, β − dp ) and ϵ > d

p , there exists a Cδ(Rd; V)-modification of r−(l+ϵ)1 X and

a constant N = N(d, p, δ, ϵ) such that

E[|r−(l+ϵ)

1 X|pδ]≤ NN.

Proof. Fix δ ∈ (0, β − dp ) and ϵ > d

p . Owing to (2.5.1), there is a constant N = N(d, p, N

2.5. Appendix 35

, δ, ϵ) such that ∫Rd

E[|r1(x)−(l+ϵ)X(x)|pV

]dx ≤ N

∫Rd

r1(x)−pϵdx ≤ NN.

Appealing to (2.5.2) and (2.3.19), we find that there is a constant N = N(d, p, δ, ϵ) suchthat ∫

|x−y|<1

E[|r1(x)−(l+ϵ)X(x) − r1(y)−(l+ϵ)X(y)|pV

]|x − y|2d+δp dxdy

≤ N∫|x−y|<1


|x − y|2d−(β−δ)p dxdy + N∫|x−y|<1

r1(y)pl|r1(x)−(l+ϵ) − r1(y)−(l+ϵ)|p

|x − y|2d+δp dxdy

≤ NN + NN∫|x−y|<1

r1(x)−p(1+ϵ) + r1(y)−p(1+ϵ)

|x − y|2d−(1−δ)p dxdy ≤ NN.

Therefore, E[r−(l+ϵ)1 X]p

δ,p ≤ NN, and hence, by Corollary 2.5.3, r−(l+ϵ)1 X has a Cδ(Rd; V)-

modification and the estimate follows immediately.

Corollary 2.5.4. Let (X(n))n∈N be a sequence of V-valued random fields defined on Rd.Assume that for some p ≥ 1, l ≥ 0 and β ∈ (0, 1], with βp > d there is a constant N > 0such that for all x, y ∈ Rd and n ∈ N,

E[|X(n)(x)|pV

]≤ Nr1(x)lp

andE

[|X(n)(x) − X(n)(y)|pV

]≤ N(r1(x)lp + r1(y)lp)|x − y|βp.

Moreover, assume that for all x ∈ Rd, limn→∞ E[|X(n)(x)|p

]= 0. Then for all δ ∈ (0, β − d

p )and ϵ > d

p ,

limn→∞

E[|r−(l+ϵ)

1 X(n)|pδ

]= 0.

Proof. Fix δ ∈ (0, β− dp ) and ϵ > d

p . Using the Lebesgue dominated convergence theorem,we attain

limn→∞

∫Rd

E[|r1(x)−(l+ϵ)X(n)(x)|pV

]dx = 0,

and therefore for all ζ ∈ (0, 1),

limn

∫ζ<|x−y|<1

E[|r1(x)−(l+ϵ)Xn(x) − r1(y)−(l+ϵ)Xn(y)|pV

]|x − y|2d+δp dxdy = 0.

2.5. Appendix 36

By repeating the proof of Corollary 2.5.3, we obtain that there is a constant N such that

∫|x−y|≤ζ

E[|r1(x)−(l+ϵ)X(n)(x) − r1(y)−(l+ϵ)X(n)(y)|pV

]|x − y|2d+δp dxdy

≤ N∫|x−y|≤ζ


|x − y|2d+(δ−β)p dxdy + N∫|x−y|≤ζ

r1(x)−p(1+ϵ) + r1(y)−p(1+ϵ)

|x − y|2d+(δ−1)p dxdy

≤ Nζβp−δp−d.

Therefore, limn→∞ E[[r−(l+ϵ)

1 X]pδ,p

]= 0, and the statement is confirmed.

Chapter 3The method of stochastic characteristicsfor parabolic SIDEs

3.1 Introduction

Let (Ω,F ,P) be a complete probability space and F0 be a sub-sigma-algebra of F . Weassume that this probability space supports a sequence w1;ϱ

t , t ≥ 0, ϱ ∈ N, of independentone-dimensional Wiener processes and a Poisson random measure p1(dt, dz) on (R+ × Z1,

B(R+)⊗Z1) with intensity measure π1(dz)dt, where (Z1,Z1, π1) is a sigma-finite measurespace. We also assume that (w1;ϱ

t )ϱ∈N and p1(dt, dz) are independent of F0. Let F = (Ft)t≥0

be the standard augmentation of the filtration (Ft)t≥0, where for each t ≥ 0,

Ft = σ(F0, (w1;ϱ

s )ϱ∈N, p1([0, s],Γ) : s ≤ t, Γ ∈ Z1).

For each real number T > 0, we let RT , OT , and PT be the F-progressive, F-optional, andF-predictable sigma-algebra onΩ×[0,T ], respectively. Denote by q1(dt, dz) = p1(dt, dz)−π1(dz)dt the compensated Poisson random measure. Let D1, E1,V1 ∈ Z be disjoint Z1-measurable subsets such that D1 ∪ E1 ∪ V1 = Z1 and π(V1) < ∞. Let (Z2,Z2, π2) be asigma-finite measure space and D2, E2 ∈ Z2 be disjoint Z2-measurable subsets such thatD2 ∪ E2 = Z2.

Fix an arbitrary positive real number T > 0 and integers d1, d2 ≥ 1. Let α ∈ (0, 2] andτ ≤ T be a stopping time. Let Fτ be the stopping time sigma-algebra associated with τand let φ : Ω × Rd1 → Rd2 be Fτ ⊗ B(Rd1)-measurable. We consider the system of SIDEson [0,T ] × Rd1 given by

dult =

((L1;l

t +L2;lt )ut + 1[1,2](α)bi

t∂iult + cll

t ult + f l

t

)dt +

(N

1;lϱt ut + glϱ

t

)dw1;ϱ

t

+

∫Z1

(I

1;lt,z ut− + hl

t(z))

[1D1(z)q1(dt, dz) + 1E1∪V1(z)p1(dt, dz)], τ ≤ t ≤ T,

ult = φ

l, t ≤ τ, l ∈ 1, . . . , d2, (3.1.1)


Lk;lt ϕ(x) : = 12(α)

12σ

k;iϱt (x)σk; jϱ

t (x)∂i jϕl(x) + 12(α)σk;iϱ

t (x)υk;llϱt (x)∂iϕ

l(x)

37

3.1. Introduction 38

+

∫Dkρk;ll

t (x, z)(ϕl(x + Hk

t (x, z)) − ϕl(x))πk(dz)

+

∫Dk

(ϕl(x + Hk

t (x, z)) − ϕl(x) − 1(1,2](α)Hk;it (x, z)∂iϕ

l(x))πk(dz)

+ 12(k)∫

E2

((Ill

d2+ ρ2;ll

t (x, z))ϕl(x + H2t (x, z)) − ϕl(x)

)π2(dz),

N1;lϱt ϕ(x) : = 12(α)σ1;iϱ

t (x)∂iϕl(x) + υ1;llϱ

t (x)ϕl(x), ϱ ∈ N,

I1;lt,zϕ(x) : = (Ill

d2+ ρ1;ll

t (x, z))ϕl(x + H1t (x, z)) − ϕl(x),

and ∫Dk

(|Hk

t (x, z)|α + |ρkt (x, z)|2

)πk(dz) +

∫Ek

(|Hk

t (x, z)|1∧α + |ρkt (x, z)|

)πk(dz) < ∞.

The summation convention with respect to repeated indices i, j ∈ 1, . . . , d1,l ∈ 1, . . . ,d2, and ϱ ∈ N is used here and below. The d2 × d2 dimensional identity matrix is denotedby Id2 . For a subset A of a larger set X, 1A denotes the 0, 1-valued function taking thevalue 1 on the set A and 0 on the complement of A. We assume that for each k ∈ 1, 2,

σkt (x) = (σk;iϱ

t (ω, x))1≤i≤d1, ϱ∈N, bt(x) = (bit(ω, x))1≤i≤d1 , ct(x) = (cll

t (ω, x))1≤l,l≤d2 ,


t (ω, x))1≤l,l≤d2, ϱ∈N, ft(x) = ( f it (ω, x))1≤i≤d2 , gt(x) = (giϱ

t (ω, x))1≤i≤d2, ϱ∈N,

are random fields on Ω× [0,T ]×Rd1 that are RT ⊗B(Rd1)-measurable. For each k ∈ 1, 2,we assume that

Hkt (x, z) = (Hk;i

t (ω, x, z))1≤i≤d1 , ρkt (x, z) = (ρk;ll

t (ω, x, z))1≤l,l≤d2 ,

are random fields on Ω × [0,T ] × Rd1 × Zk that are PT ⊗ B(Rd1) ⊗Zk-measurable. More-over, we assume that ht(x, z) = (hi

t(ω, x, z))1≤i≤d2 is a random field on Ω × [0,T ] × Rd1 thatis PT ⊗ B(Rd1)-measurable. Systems of linear SIDEs appear in many contexts. Theymay be considered as extensions of both first-order symmetric hyperbolic systems and lin-ear fractional advection-diffusion equations. The equation (3.1.1) also arises in non-linearfiltering of semimartingales as the equation for the unormalized filter of the signal (see,e.g., [Gri76] and [GM11]). Moreover, (3.1.1) is intimately related to linear transforma-tions of inverse flows of jump SDEs and it is precisely this connection that we will exploitto obtain solutions.

Systems of linear SIDEs appear in many contexts. They may be considered as ex-tensions of both first-order symmetric hyperbolic systems and linear fractional advec-tion-diffusion equations. The equation (3.1.1) also arises in non-linear filtering of semi-martingales as the equation for the unormalized filter of the signal (see, e.g., [Gri76] and


[GM11]). Moreover, (3.1.1) is intimately related to linear transformations of inverse flowsof jump SDEs and it is precisely this connection that we will exploit to obtain solutions.

There are various techniques available to derive the existence and uniqueness of clas-sical solutions of linear parabolic SPDEs and SIDEs. One approach is to develop a theoryof weak solutions for the equations (e.g. variational, mild solution, or etc...) and thenstudy further regularity in classical function spaces via an embedding theorem. We referthe reader to [Par72, Par75, MP76, KR77, Tin77a, Gyö82, Wal86, DPZ92, Kry99, CK10,PZ07, Hau05, RZ07, BvNVW08, HØUZ10, LM14b] for more information about weaksolutions of SPDEs driven by continuous and discontinuous martingales and martingalemeasures. This approach is especially important in the non-degenerate setting where somesmoothing occurs and has the obvious advantage that it is broader in scope. Another ap-proach is to regard the solution as a function with values in a probability space and usethe method of deterministic PDEs (i.e. Schauder estimates, see, e.g. [Mik00, MP09]).A third approach is a direct one that uses solutions of stochastic differential equations.The direct method allows to obtain classical solutions in the entire Hölder scale whilenot restricting to integer derivative assumptions for the coefficients and data. In thischapter, we derive the existence of a classical solutions of (3.1.1) with regular coefficientsusing a Feynman-Kac-type transformation and the interlacing of the space-inverse (firstintegrals [KR81]) of a stochastic flow associated with the equation. The construction ofthe solution gives an insight into the structure of the solution as well. We prove that thesolution of (3.1.1) is unique in the class of classical solutions with polynomial growth (i.e.weighted Hölder spaces). As an immediate corollary of our main result, we obtain the ex-istence and uniqueness of classical solutions of linear partial integro-differential equations(PIDEs) with random coefficients, since the coefficients σ1, H1, a1, ρ1, and free terms gand h can be zero. Our work here directly extends the method of characteristics for de-terministic first-order PDEs and the well-known Feynman-Kac formula for deterministicsecond-order PDEs.

In the continuous case (i.e. H1 ≡ 0,H2 ≡ 0, h ≡ 0), the classical solution of (3.1.1)was constructed in [KR81, Kun81, Kun86a, Roz90] (see references therein as well) us-ing the first integrals of the associated backward SDE. This method was also used toobtain classical solutions of (3.1.1) in [DPMT07]. In the references above, the forwardLiouville equation for the first integrals of associated stochastic flow was derived directly.However, since the backward equation involves a time reversal, the coefficients and in-put functions are assumed to be non-random. The generalized solutions of (3.1.1) withd2 = 1, non-random coefficients, non-degenerate diffusion, and finite measures π1 = π2

were discussed in [MB07]. In this chapter, we give a direct derivation of (3.1.1) and allthe equations considered are forward, possibly degenerate, and the coefficients and inputfunctions are adapted. For other interesting and related developments, we refer the reader


to [Pri12, Zha13, Pri14], which all concern the fascinating regularizing property of noisein the case of non-degenerate noise.

This chapter is organized as follows. In Section 3.2, our notation is set forth and themain results are stated. In Section 3.3, the main theorems are proved. In Section 3.4, theappendix, auxiliary facts used throughout the chapter are discussed.


In this chapter, elements of Rd1 and Rd2 are understood as column vectors and elements ofRd2

1 and Rd22 are understood as matrices of dimension d1 × d1 and d2 × d2, respectively. We

also adopt the notation of Chapter 2. If we do not specify to which space the parametersω, t, x, y, z and n belong, then we mean ω ∈ Ω, t ∈ [0,T ], x, y ∈ Rd1 , z ∈ Zk, and n ∈ N.

Let us introduce some regularity conditions on the coefficients and free terms. Weconsider these assumptions for β > 1 ∨ α and β > α.

Assumption 3.2.1 (β). (1) There is a constant N0 > 0 such that for each k ∈ 1, 2 and allω, t ∈ Ω × [0,T ],

|r−11 bt|0 + |∇bt|β−1 + |r−1

1 σkt |0 + |∇σ

kt |β−1 ≤ N0.

Moreover, for each k ∈ 1, 2 and all (ω, t, z) ∈ Ω × [0,T ] × (Dk ∪ Ek),

|r−11 Hk

t (z)|0 ≤ Kkt (z) and |∇Hk

t (z)|β−1 ≤ Kkt (z)

where Kk, Kk : Ω × [0,T ] × (Dk ∪ Ek) → R+ are PT ⊗ Zk-measurable functions

satisfying

Kkt (z) + Kk

t (z) +∫

Dk

(Kk

t (z)α + Kkt (z)2

)πk(dz) +

∫Ek

(Kk

t (z)1∧α + Kkt (z)

)πk(dz) ≤ N0,

for all (ω, t, z) ∈ Ω × [0,T ] × (Dk ∪ Ek).(2) There is a constant η ∈ (0, 1) such that for all (ω, t, x, z) ∈ (ω, t, x, z) ∈ Ω × [0,T ] ×

Rd1 × (Dk ∪ Ek) : |∇Hkt (ω, x, z)| > η,∣∣∣∣(Id1 + ∇Hk

t (x, z))−1∣∣∣∣ ≤ N0.

Assumption 3.2.2 (β). There is a constant N0 > 0 such that for each k ∈ 1, 2 and all(ω, t) ∈ Ω × [0,T ],

|ct|β + |υkt |β + |r

−θ1 ft|β + |r

−θ1 gt|β ≤ N0.


Moreover, for each k ∈ 1, 2 and all (ω, t, z) ∈ Ω × [0,T ] × (Dk ∪ Ek),

|ρkt (z)|β ≤ lk

t (z), |r−θ1 ht(z)|β ≤ lkt (z),

where lk : Ω × [0,T ] × Zk → R+ are PT ⊗Zk-measurable function satisfying

lkt (z) +

∫Dk

lkt (z)2πk(dz) +

∫Ek

lkt (z)πk(dz) ≤ N0,

for all (ω, t, z) ∈ Ω × [0,T ] × (Dk ∪ Ek).

Remark 3.2.1. It follows from Lemma 3.4.10 and Remark 3.4.11 that if Assumption3.2.1(β) holds for some β > 1 ∨ α, then for all ω, t, and z ∈ Dk ∪ Ek, x 7→ Hk

t (x, z) :=x + Hk

t (x, z) is a diffeomorphism.

Let Assumptions 3.2.1(β) and 3.2.2(β) hold for some β > 1 ∨ α and β > α. In ourderivation of a solution of (3.1.1), we first obtain a solution of an equation of a specialform. Specifically, consider the system of SIDEs on [0,T ] × Rd1 given by

dult =

((L1;l

t +L2;lt )ut + bi

t∂iult + cll

t ult + f l

t

)dt +

(N

1;lϱt ut + glϱ

t

)dw1;ϱ

t

+

∫Z1

(I

1;lt,z ut− + hl

t(z))

[1D1(z)q1(dt, dz) + 1E1(z)p1(dt, dz)], τ < t ≤ T,

ult = φ

l, t ≤ τ, l ∈ 1, . . . , d2, (3.2.1)

where

bit(x) : = 1[1,2](α)bi

t(x) +2∑

k=1

12(α)σk; jϱt (x)∂ jσ

k;iϱt (x)

+

2∑k=1

1(1,2](α)∫

Dk

(Hk;i

t (x, z) − Hk;it (Hk;−1

t (x, z), z))πk(dz),

cllt (x) : = cll

t (x) +2∑

k=1

12(α)σk; jϱt (x)∂ jυ

k;llϱt (x)

+

2∑k=1

∫Dk

(ρk;llt (x, z) − ρk;ll

t (Hk;−1t (x, z), z))πk(dz),

f lt (x) : = f l

t (x) + σ1; jϱt (x)∂ jg

lϱt (x) +

∫D1

(hl

t(x, z) − hlt(H

1;−1t (x, z), z)

)π1(dz).

Let (w2;ϱt )ϱ∈N, t ≥ 0, ϱ ∈ N, be a sequence of independent one-dimensional Wiener

processes. Let p2(dt, dz) be a Poisson random measure on ([0,∞) × Z2,B([0,∞) ⊗ Z2)with intensity measure π2(dz)dt. Extending the probability space if necessary, we take w2


and p2(dt, dz) to be independent of w1 and p1(dt, dz). Let

Ft = σ((w2;ϱ

s )ϱ∈N, p2([0, s],Γ) : s ≤ t, Γ ∈ Z2)

and F =(Ft

)t≤T

be the standard augmentation of(Ft ∨ Ft

)t≤T

. Denote by q2(dt, dz) =p2(dt, dz) − π2(dz)dt the compensated Poisson random measure. We associate with theSIDE (3.2.1), the F-adapted stochastic flow Xt = Xt(x) = Xt(τ, x), (t, x) ∈ [0,T ] × Rd1 ,generated by the SDE

dXt = −1[1,2](α)bt(Xt)dt +2∑

k=1

12(α)σk;ϱt (Xt)dwk;ϱ

t

−

2∑k=1

∫Dk

Hkt (Hk;−1

t (Xt−, z), z)[pk(dt, dz) − 1(1,2](α)πk(dz)dt]

−

2∑k=1

∫Ek

Hkt (Hk;−1

t (Xt−, z), z)pk(dt, dz), τ < t ≤ T,

Xt = x, t ≤ τ, (3.2.2)

and the F-adapted random field Φt(x) = Φt(τ, x), (t, x) ∈ [0,T ] × Rd1 , solving the linearSDE given by

dΦt(x) = (ct(Xt(x))Φt(x) + ft(Xt(x))) dt +2∑

k=1

υk;ϱt (Xt(x))Φt(x)dwk;ϱ

t + gϱt (Xt(x))dw1;ϱt

+

2∑k=1

∫Zkρk

t (Hk;−1t (Xt−(x), z), z)Φt−(x)[1Dk(z)qk(dt, dz) + 1Ek(z)pk(dt, dz)]

+

∫Z1

ht(H1;−1t (Xt−(x), z), z)[1D1(z)q1(dt, dz) + 1E1(z)p1(dt, dz)], τ < t ≤ T,

Φt(x) = φ(x), t ≤ τ.

The coming theorem is our existence, uniqueness, and representation theorem for(3.2.1). Let us describe our solution class. For each β′ ∈ (0,∞), denote by Cβ

′

(Rd1; Rd2)the linear space of all F-adapted random fields v = vt(x) such that P-a.s.

1[τn,τn+1)r−λn1 v ∈ D([0,T ];Cβ

′

(Rd1 ,Rd2)),

where (τn)n≥0 is an increasing sequence of F-stopping times with τ0 = 0 and τn = T forsufficiently large n, and where for each n, λn is a positive Fτn-measurable random variable.

Theorem 3.2.2. Let Assumptions 3.2.1(β) and 3.2.2(β) hold for some β > 1∨α and β > α.For any stopping time τ ≤ T and Fτ ⊗ B(Rd1)-measurable random field φ such that for


some β′ ∈ (α, β ∧ β) and θ′ ≥ 0, P-a.s. r−θ′

1 φ ∈ Cβ′

(Rd1; Rd2), there exists a unique solutionu = u(τ) of (3.2.1) in Cβ

′

(Rd1; Rd2) and for all (t, x) ∈ [0,T ] × Rd1 , P-a.s.

ut(τ, x) = E[Φt(τ, X−1

t (τ, x))|Ft

]. (3.2.3)

Moreover, for all ϵ > 0 and p ≥ 2, there is a constant N = N(d1, d2, p,N0,T, β′, η1, η2, ϵ,

θ, θ′) such that

E[supt≤T|r−θ∨θ

′−ϵ1 ut(τ)|pβ′

∣∣∣Fτ] ≤ N(|r−θ′

1 φ|pβ′ + 1). (3.2.4)

Using Itô’s formula, it is easy to verify that if m = 1 and

gt(x) = 0, ht(x) = 0, and ρkt (x, z) ≥ −1,

for all (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × (Dk ∪ Ek), k ∈ 1, 2, then

Φt(x) = Ψt(x)ϕ(x) + Ψt(x)∫

]τ,τ∨t]Ψ−1

s (x) fs(Xs(x))ds, (3.2.5)

where P-a.s. for all t and x, Ψt(x) := eBt(x), and

Bt(x) :=∫

[τ,τ∨t]

cs(Xs(x)) −2∑

k=1

12υk;ϱ

s (Xs(x))υk;ϱs (Xs(x))

ds

+

2∑k=1

∫]τ,τ∨t]

υk;ϱs (Xs(x))dwk;ϱ

s

−

2∑k=1

∫]τ,τ∨t]

∫Dk

(ln

(1 + ρk

s(Hk;−1s (Xs−(x), z), z)

)− ρk

s(Hk;−1s (Xs−(x), z), z)

)πk(dz)ds

2∑k=1

∫]τ,τ∨t]

∫Zk

ln(1 + ρk

s(Hk;−1s (Xs−(x), z), z)

)[1Dk(z)qk(ds, dz) + 1Ek(z)pk(ds, dz)].

The following corollary then follows directly from (3.2.3) and (3.2.5).

Corollary 3.2.3. Let m = 1 and assume that for each k ∈ 1, 2 and all (ω, t, x, z) ∈Ω × [0,T ] × Rd1 × (Dk ∪ Ek),

gt(x) = 0, ht(x, z) = 0, ρkt (x, z) ≥ −1.

Moreover, let Assumptions 3.2.1(β) and 3.2.2(β) hold for some β > 1 ∨ α and β > α. Letτ ≤ T be a stopping time and φ be a Fτ ⊗ B(Rd1)-measurable random field such that forsome β′ ∈ (α, β ∧ β) and θ′ ≥ 0, P-a.s. r−θ

′

1 φ ∈ Cβ′

(Rd1; Rd2).

(1) If for all (ω, t, x) ∈ Ω × [0,T ] × Rd1 , ft(x) ≥ 0 and φ(x) ≥ 0, then the solution u of(3.1.1) satisfies ut(x) ≥ 0, P-a.s. for all (t, x) ∈ [0,T ] × Rd1 .


(2) If for all (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × (Dk ∪ Ek), k ∈ 1, 2, υkt (x) = 0, ft(x) ≤ 0,

ct(x) ≤ 0, φ(x) ≤ 1, and ρkt (x, z) ≤ 0, then the solution u of (3.1.1) satisfies ut(x) ≤ 1,

P-a.s. for all (t, x) ∈ [0,T ] × Rd1 .

Remark 3.2.4. Since L2 can be the zero operator, both Theorem 3.2.2 and Corollary 3.2.3apply to fully degenerate SIDEs and PIDEs with random coefficients.

Now, let us discuss our existence and uniqueness theorem for (3.1.1). We constructthe solution of u = u(τ) of (3.1.1) by interlacing the solutions of (3.2.1) along a sequenceof large jump moments (see Section 3.3.5). By using an interlacing procedure we arealso able to drop the condition of boundedness of (I + ∇H1

t (x, z))−1 on the set (ω, t, x, z) ∈(ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × (D1 ∪ E1) : |∇H1

t (ω, x, z)| > ηk. Also, in order to removethe terms in b, c, and f that appear in (3.2.1), but not in (3.1.1), we subtract terms fromthe relevant coefficients in the flow and the transformation. However, in order to do this,we need to impose stronger regularity assumptions on some of the coefficients and freeterms. We will introduce parameters µ1, µ2, δ1, δ2 ∈ [0, α2 ] to the regularity assumptions.These parameters allows for a trade-off of integrability in z and regularity in x of thecoefficients Hk

t (x, z), ρkt (x, z), hk

t (x, z). It is worth mentioning that the removal of terms andthe interlacing procedure are independent of each other and that it is due only to the weakassumptions on H1 and ρ1 on the set V1 that we do not have moment estimates and asimple representation property like (3.2.4) for the solution of (3.1.1). However, there is arepresentation of the solution and we refer the reader to the proof of the coming theoremfor the description of the representation.

We introduce the following assumptions for β > 1∨α and β > α. For each k ∈ 1, 2, letDk be the trace sigma-algebra of Dk ∪Ek relative toZk andV1 be the trace sigma-algebraof V1 relative toZ1.

Assumption 3.2.3 (β). (1) There is a constant N0 > 0 such that for each k ∈ 1, 2 and all(ω, t) ∈ Ω × [0,T ],

|r−11 bt|0 + |∇bt|β−1 + |σ

kt |β+1 ≤ N0.

(2) For each k ∈ 1, 2 , there are real-numbers δk, µk ∈ [0, α2 ] such that for all (ω, t) ∈Ω × [0,T ],

|υkt |β+1 ≤ N0, if σk

t , 0, |gt|β+1 ≤ N0, if σ1t , 0,

|Hkt (z)|0 ≤ Kk

t (z), |∇Hkt (z)|β+δk−1 ≤ Kk

t (z), ∀z ∈ Dk,∑|γ|=[β+δk]−

[∂γHkt (z)]β+δk+ ≤ Kk

t (z), ∀z ∈ Dk,

|r−11 Hk

t (z)|0 ≤ Kkt (z), |∇Hk

t (z)|β−1 ≤ Kkt (z), ∀z ∈ Ek,


|ρkt (z)|β+µk ≤ lk

t (z),∑

|γ|=[β+µk]−

[∂γρkt (z)]β+µk+ ≤ lk

t (z), ∀z ∈ Dk,

|r−θ1 ht(z)|β+µ1 ≤ l1t (z),

∑|γ|=[β+µ1]−

[∂γht(z)]β+µ1+ , ∀z ∈ D1,

where Kk, Kk, Kk, lk, lk : Ω×[0,T ]×(Dk∪Ek)→ R+ arePT⊗Dk-measurable functions

satisfying for all (ω, t, z) ∈ Ω × [0,T ],

Kkt (z) + Kk

t (z) + lkt (z) + lk

t (z) + lkt (z) ≤ N0, ∀z ∈ Dk ∪ Ek,∫

Dk

(Kk

t (z)α + Kkt (z)2 + Kk

t (z)α

α−δk + Kkt (z)2 + lk

t (z)2 + lkt (z)

α

α−µk + lkt (z)2

)πk ≤ N0(dz),∫

Ek

(Kk

t (z)1∧α + Kkt (z)

)πk(dz) ≤ N0.

(3) There is a constant η ∈ (0, 1) such that for all (ω, t, x, z) ∈ (ω, t, x, z) ∈ Ω × [0,T ] ×Rd1 × Z2 : |∇H2

t (ω, x, z)| > η,∣∣∣∣(Id1 + ∇H2t (x, z)

)−1∣∣∣∣ ≤ N0.

Assumption 3.2.4 (β). (1) There is a constant N0 > 0 such that for each k ∈ 1, 2 and all(ω, t) ∈ Ω × [0,T ],

|ct|β + |r−θ1 ft|β ≤ N0,

|υkt |β ≤ N0, if σk

t = 0, |gt|β ≤ N0, if σ1t = 0,

|ρk(t, z)|β ≤ lkt (z), ∀z ∈ Ek, |r−θ1 ht(z)|β ≤ l1

t (z), ∀z ∈ E1,

where for all (ω, t) ∈ Ω × [0,T ],∫

Ek lkt (z)πk(dz) ≤ N0.

(2) There exist processes ξ, ζ : Ω × [0,T ] × V1 → R+ that are PT ⊗ V1measurable and

satisfy|r−ξt(z)

1 H1t (z)|β∨1 + |r

−ξt(z)1 ρ1

t (z)|β + |r−ξt(z)1 ht(z)|β ≤ ζt(z),

for all (ω, t, z) ∈ Ω × [0,T ] × V1.

We now state our existence and uniqueness theorem for (3.1.1).

Theorem 3.2.5. Let Assumptions 3.2.3(β) and 3.2.4(β) hold for some β > 1∨α and β > α.For any stopping time τ ≤ T and Fτ ⊗ B(Rd1)-measurable random field φ such that forsome β′ ∈ (α, β ∧ β) and θ′ ≥ 0, P-a.s. r−θ

′

1 φ ∈ Cβ′

(Rd1; Rd2), there exists a unique solutionu = u(τ) of (3.1.1) in Cβ

′

(Rd1; Rd2).

3.3. Proof of main theorems 46

3.3 Proof of main theorems

We will first prove uniqueness of the solution of (3.2.1) in the class Cβ′

(Rd1; Rd2). Theexistence part of the proof of Theorem 3.2.2 is divided into a series of steps. In the firststep, by appealing to the representation theorem we derived for solutions of continuousSPDEs shown in Theorem 2.4 in [LM14c], we use an interlacing procedure and the stronglimit theorem given in Theorem 2.3 in [LM14c] to show that the space inverse of the flowgenerated by a jump SDE (i.e. the SDE (3.2.2) without the uncorrelated noise) solves adegenerate linear SIDE. Then we linearly transform the inverse flow of a jump SDE toobtain solutions of degenerate linear SIDEs with free and zero-order terms and an initialcondition. In the last step of the proof of Theorem 3.2.2, we introduce an independentWiener process and Poisson random measure as explained above, apply the results weknow for fully degenerate equations, and then take the optional projection. In the lastsection, Subsection 3.3.4, we prove Theorem 3.2.5 using an interlacing procedure andremoving the extra terms in b, c and f . The uniqueness of the solution u of (3.1.1) followsdirectly from our construction.

3.3.1 Proof of uniqueness for Theorem 3.2.2

Proof of Uniqueness for Theorem 3.2.2. Fix a stopping time τ ≤ T and Fτ ⊗ B(Rd1)-measurable random field φ such that for some β′ ∈ (α, β ∧ β) and θ′ ≥ 0, P-a.s. r−θ

′

1 φ ∈

Cβ′

(Rd1; Rd2). In this section, we will drop the dependence of processes on t, x, and z whenwe feel it will not obscure the argument. Let u1(τ) and u2(τ) be solutions of (3.2.1) in Cβ

′

.It follows that v := u1(τ) − u2(τ) solves

dvlt = [(L1;l

t +L2;lt )vt + bi

t∂ivlt + cll

t vlt]dt +N1;ϱ

t vltdw1;ϱ

t

+

∫Z1I

1;lt,z vt−[1D1(z)q1(dt, dz) + 1E1(z)p1(dt, dz)], τ < t ≤ T,

vlt = 0, t ≤ τ, l ∈ 1, . . . , d2,

and P-a.s.1[τn,τn+1)(·)r

−λn1 v ∈ D([0,T ];Cβ

′

(Rd1 ,Rd2)),

where (τn)n≥0 is an increasing sequence of F-stopping times with τ0 = 0 and τn = T forsufficiently large n, and where for each n, λn is a positive Fτn-measurable random variable.Clearly, it suffices to take τ1 = τ and λ0 = 0. Thus, vt(x) = 0 for all (ω, t) ∈ [[τ0, τ1)).Assume that for some n, P-a.s. for all t and x, vt∧τn(x) = 0. We will show that P-a.s. for all


t and x, vt(x) := v(τn∨t)∧τn+1(x) = 0. Applying Itô’s formula, for all x, P-a.s. for all t, we find

d|vt|2 =

(2vl

tL1;lt vt + |N

1t vt|

2 + 2vltb

it∂ivl

t + 2vltc

llt vl

t

)dt

+

(2vl

tI1;lt,z vt +

∫D1∪E1

|I1;lt,z vt|

2π1(dz))

dt

+(2vl

tL2;lt vt + 2vl

tI2;lt,z vt

)dt + 2vl

tN1;ϱt vl

tdw1;ϱt

+

∫Z1

(2vl

t−I1;lt,z vt− + |I

1;lt,z vt−|

2)

q1(dt, dz), τn < t ≤ τn+1,

|vt|2 = 0, t ≤ τn, l ∈ 1, . . . , d2, (3.3.1)


Lk;lϕ :=

12σk;iϱσk; jϱ∂i jϕ

l + σk; jϱ∂ jσk;iϱ∂iϕ

l + σk;iϱυk;llϱ∂iϕl + σk; jϱ∂ jak;llϱϕl

and

Ik;lϕ : =

∫Dk

(ρk;llϕl(Hk) − ρk;ll(Hk;−1)ϕl

)πk(dz)

+

∫Dk

(ϕl(Hk) − ϕl + 1(1,2](α)Fk;i∂iϕ

l)πk(dz)

+

∫Ek

((Ill

d2+ ρk;ll)ϕl(Hk) − ϕl

)πk(dz).

For each ω and t, let

Qt =

∫Rd1|vt(x)|2r−λ1 (x)dx,

where λ = λn + (d′ + 2)/2 and d′ > d1. Note that

EQt ≤

∫Rd1

r−d′1 (x)dxE|r−λn

1 vt|0 < ∞.

It suffices to show that supt≤T EQt = 0. To this end, we will multiply the equation (3.3.1)by the weight r−2λ

1 = r−2λn+11 r−d′

1 , integrate in x, and change the order of the integrals in timeand space. Thus, we must verify the assumptions of stochastic Fubini theorem hold (seeCorollary 3.4.13 and Remark 3.4.14 as well) with the finite measure µ(dx) = r−d′

1 (x)dx onRd1 . Since b and σk have linear growth an υk and c are bounded, owing to Lemma 3.4.6,we easily obtain that there is a constant N = N(d1, d2,N0, λn) such that P-a.s for all t,

∫Rd1

2∑k=1

2|r−λn1 v||r−λn−2

1 Lkv| + |rλn−1

1 N1v|2 r−d′

1 dx ≤ N supt≤T|r−λn

1 v|2β′ ,


∫Rd1

4|r−λn1 v|2|r−λn−1

1 N1v|2r−d′1 dx ≤ N sup

t≤T|r−λn

1 vt|4β′ ,

and ∫Rd1

(2|r−λn

1 v|r−λn−11 b∂iv| + 2|r−λn

1 v||r−λn1 cv|

)r−d′


1 vt|2β′ .

For all ϕ ∈ Cαloc(R

d1; Rd2) and all k, ω, t, x, p, and z,

r−p1 (ϕ(Hk) − ϕ + 1(1,2](α)Fk;i∂iϕ)

= ϕ(Hk) − ϕ − 1(1,2](α)Hk;i∂iϕ + 1(1,2](α)(Hk;i + Fk;i)∂iϕ

+p1(1,2](α)(Hk;i + Fk;i)r−21 xiϕ +

rp1 (Hk)

rp1

− 1 (ϕ(Hk) − 1(1,2](α)ϕ)

+1(1,2](α)rp

1 (Hk)rp

1

− 1 + pHk;ir−21 xi

ϕ, (3.3.2)

where ϕ := r−pϕ. By Taylor’s formula, for all ϕ ∈ Cα(Rd1; Rd2) and all k, ω, t, x, and z, wehave

|ϕ(Hk) − ϕ − 1(1,2](α)Hk;i∂iϕ| ≤ rα1 |ϕ|α|r−11 H|α0 . (3.3.3)

Combining (3.3.2), (3.3.3), and the estimates given in Lemma 3.4.10 (1), for all k, ω, t, xand z, we obtain

r−α1 |ρk(Hk;−1) − ρk| ≤ N|ρ|α∧1|r−1

1 Hk|α∧10

and

r−λn−α1 |v(Hk) − v + 1(1,2](α)Fk;i∂iv|

≤ N |r−λn1 v|α

(|r−1

1 Hk|α0 + |r−11 H|0[Hk]1 + |r−1

1 H|[α]−+10 + [H][α]−+1

1

), (3.3.4)

for some constant N = N(d1, λn,N0, η1, η2). Therefore, P-a.s for all t,

∫Rd1

2∑k=1

2|r−λn1 v||r−λn−2

1 Ikv| +

∫D1∪E1

|r−λ−11 Izv|2π1(dz)

r−d′1 dx ≤ N sup

t≤T|r−λn

1 v|2β′ ,

and ∫Rd1

(2|r−λn

1 v||r−λn−21 Ik

z v| + |r−λn−11 Izv|2

)2r−d′


1 vt|4β′ ,

for some constant N = N(d1, d2, λn,N0, η1, η2).

Let L2(Rd1; Rd2) be the space of square-integrable functions f : Rd1 → Rd2 with norm∥ · ∥0 and inner product (·, ·)0. Moreover, let L2(Rd1; ℓ2(Rd2)) be the space of square-integrable functions f : Rd1 → ℓ2(Rd2) with norm ∥ · ∥0. With the help of the above


estimates and Corollary 3.4.13, denoting v = r−λv, P-a.s. for all t, we have

d∥vt∥20 =

(2(vl

t, L1t vt)0 + ∥N

1t vt∥

20 + 2(vt, I

1t,zvt)0 +

∫D1∪E1

∥I1t,zvt∥

20π

1(dz))

dt

+(2(vt, bi

t∂ivt + cltv

lt)0 + 2(vt, L

2t vt)0 + 2(vt, I

2t,zvt)0

)dt + 2(vt, N

1;ϱt vt)0dw1;ϱ

t

+

∫Z1

(2(vt−, I

1t,zvt−)0 + ∥I

1t,zvt−∥

20

)q1(dt, dz), τn < t ≤ τn+1,

∥vt∥20 = 0, t ≤ τn, l ∈ 1, . . . , d2, (3.3.5)

where all coefficients and operators are defined as in (3.2.1) with the following changes:

(1) for each k ∈ 1, 2, υk is replaced with

υk;ll := υk;ll + 12(α)λσk;iϱr−21 xiδll;

(2) for each k ∈ 1, 2, ρk replaced with

ρk;ll := ρk;ll +

rλ1(Hk)rλ1

− 1 (Ill

d2+ ρk;ll);

(3) c is replaced with

cll = cll + λbir−2xiδll +

2∑k=1

λ2σk;iϱσk; jϱr−41 xix j

+

2∑k=1

∫Dk

rλ1rλ1(Hk;−1)

− 1 (Ill

m + ρk(Hk;−1)) − 1(1,2](α)λr−2

1 xiHk;i(Hk;−1) πk(dz).

Since for all k, ω and t, |r−11 σk|0 + |r−1

1 ∇σk|β−1 + |υ

k|β ≤ N0, for β > 1 ∨ α and β > α, it isclear that |υk|α ≤ N. Moreover, since for all k, ω and t, |r−1

1 Hk|0 + |Hk|β ≤ Kk and |ρ|β′ ≤ lk,applying the estimates in Lemma (3.4.10) (1), we get

|ρk|α ≤ lk + Kk(1 + lk) and |c|α ≤ N0.

We will now estimate the drift terms of (3.3.5) in terms of ∥vt∥20. We write f ∼ g if∫

Rd1| f (x)|dx =

∫Rd1|g(x)|dx and f ≪ g if

∫Rd1| f (x)|dx ≤

∫Rd1|g(x)|dx. Using the divergence

theorem, for any v : Rd1 → Rd2 , σ : Rd1 → Rd1 and υ : Rd1 → R2d2 and all x, we get

σiσ jvlvli j ∼

12

(σiσ j)i jv − σiσ jvliv

lj = (σi

i jσj + σi

jσji )|v|

2 − σiσ jvliv

lj,

2σijσ

jvlvli ∼ −(σi

jσj)i|v|2 = (σi

i jσj + σi

jσji )|v|

2,


and

σivlυllvli + σ

ivlυllvli = σ

ivlυllsymvl

i ∼ −(σiυllsym)i|v|2 = −(σi

iυllsym + σ

iυllsym)|v|2,

where υllsym = (υll + υll)/2. Consequently, for all ω, t, and x, we have

2vlL

1;lv + |N1v|2 ∼12

(| divσ1|2 − ∂iσ

1; jϱ∂ jσ1;iϱ

)|v|2

− υ1;llϱsym vlvl divσ1;ϱ + |υ1v|2 ≪ N |v|2

and2vlL

(2);lv ≪ −(1 + ϵ)|σ2;i∂iv|2 + N |v|2,

for any ϵ > 0, where in the last estimate we have also used Young’s inequality. By Lemma3.4.10 (2) and basic properties of the determinant, there is a constant N = N(d,N0, η

1, η2)such that for all k, ω, t, x, and z,

det Hk;−1 − 1 = det(Id1 + Fk) − 1 ≤ |∇Fk| ≤ N |∇Hk|

anddet Hk;−1 − 1 − div Fk ≤ |∇Fk|2 ≤ N|∇Hk|2.

Thus, integrating by parts, for all ω, t, and x, we get

2vlI

1;lv +∫

D1∪E1|I1v|2π1(dz) ∼ 2

∫D1ρ1;ll

sym(H1;−1)(det∇H1;−1 − 1)π1(dz)vlvl

+

∫D1∪E1

(det∇H1;−1 − 1 + 1(1,2](α)1D1 div F1

)π1(dz)|v|2

+

∫D1∪E1

(1E12ρ1;ll

sym(H1;−1)vlvl + |ρ1(H1;−1)v|2)

det∇H1;−1π1(dz)

≪ N(∫

D1

(K1(z)2 + l1(z)K1(z) + l1(z)2

)π1(dz) +

∫E1

(Kk(z) + lk(z)

)π1(dz)

)|v|2.

Analogously, for all ω, t, and x, we attain

2vlI

2;lv ≤ −(1 + ϵ)∫

D2∪E2|v(H2) − v|2π2(dz) + N|v|2.

Therefore, combining the above estimates, P-a.s. for all t,

Qt ≤ N∫ t

0Qsds + Mt, (3.3.6)

where (Mt)t≤T is a càdlàg square-integrable martingale. Taking the expectation of (3.3.6)


and applying Gronwall’s lemma, we get supt≤T EQt = 0, which implies that P-a.s. for all tand x, vt(x) = 0. This completes the proof.

3.3.2 Small jump case

Set (wϱ)ϱ∈N = (w1;ϱ)ϱ∈N, (Z,Z, π) = (Z1,Z1, π1), p(dt, dz) = p1(dt, dz), and q(dt, dz) =q1(dt, dz). Let σt(x) = (σiϱ

t (x))1≤i≤d1,ϱ≥1 be a ℓ2(Rd1)-valued RT ⊗B(Rd1)-measurable func-tion defined on Ω × [0,T ] × Rd1 and Ht(x, z) = (Hi

t(x, z))1≤i≤d1 be a PT ⊗ B(Rd1) ⊗ Z-measurable function defined on Ω × [0,T ] × Rd1 × Z.

We introduce the following assumption for β > 1 ∨ α.

Assumption 3.3.1 (β). (1) There is a constant N0 > 0 such that for all (ω, t) ∈ Ω× [0,T ],

|r−11 bt|0 + |r−1

1 σt|0 + |∇bt|β−1 + |∇σt|β−1 ≤ N0.

Moreover, for all (ω, t, z) ∈ Ω × [0,T ] × Z,

|r−11 Ht(z)|0 ≤ Kt(z) and |∇Ht(z)|β−1 ≤ Kt(z),


Kt(z) + Kt(z) +∫

Z

(Kt(z)α + Kt(z)2

)π(dz) ≤ N0,

for all (ω, t, z) ∈ Ω × [0,T ] × Z.(2) There is a constant η ∈ (0, 1) such that for all (ω, t, x, z) ∈ (ω, t, x, z) ∈ Ω × [0,T ] ×

Rd1 × Z : |∇Ht(ω, x, z)| > η,

|(Id1 + ∇Ht(x, z)

)−1| ≤ N0.

Let Assumption 3.3.1(β) hold for some β > 1 ∨ α. Let τ ≤ T be a stopping time.Consider the system of SIDEs on [0,T ] × Rd1 given by

dvt(x) =(12(α)

12σ

iϱt (x)σ jϱ

t (x)∂i jvt(x) + bit(x)∂ivt(x))

dt + 12(α)σiϱt (x)∂ivt(x)dwϱ

t

+

∫Z

(vt−(x + Ht(x, z)) − vt−(x)) [1(1,2](α)q(dt, dz) + 1[0,1](α)p(dt, dz)]

+ 1(1,2](α)∫

Z(vt(x + Ht(x, z)) − vt(x) + Ft(x, z)∂ivt(x)) π(dz)dt, τ < t ≤ T,

vt(x) = x, t ≤ τ, (3.3.7)


whereb

it(x) := 1[1,2](α)bi

t(x) + 12(α)σ jϱt (x)∂ jσ

iϱt (x)

andFt(x, z) := −Ht(H−1

t (x, z), z).

We associate with (3.3.7) the stochastic flow Yt = Yt(τ, x), (t, x) ∈ [0,T ] × Rd1 , generatedby the SDE

dYt = −1[1,2](α)bt(Yt)dt − 12(α)σϱt (Yt)dwϱ

t

+

∫Z

Ft(Yt−, z)[1(1,2](z)q(dt, dz) + 1[0,1](z)p(dt, dz)], τ < t ≤ T, (3.3.8)

Yt = x, t ≤ τ.

Owing to parts (1) and (2) of Lemma 3.4.10, for all ω, t, and z, the inverse of the mappingFt(x, z) := x+Ft(x, z) = x−Ht(H−1

t (x, z), z) is Ht(x, z) := x+Ht(x, z) and there is a constantN = N(d1,N0, β, η) such that for all ω, t, x, y, and z,

|r−11 Ft(z)|0 ≤ NKt(z), |∇Ft(z)|β−1 ≤ Kt(z), |(Id1 + ∇Ft(x, z))−1| ≤ N.

Thus, by Theorem 2.1 in [LM14c], there is a modification of the solution of (3.3.8), whichwe still denote by Yt = Yt(τ, x), that is a Cβ

′

loc-diffeomorphism for any β′ ∈ [1, β). Moreover,P-a.s. Y·(τ, ·),Y−1

· (τ, ·) ∈ D([0,T ];Cβ′

loc(Rd1; Rd1)), and Y−1

t− (τ, ·) coincides with the inverseof Yt−(τ, ·) for all t. In the proof of the following proposition, we will show that the inverseflow Y−1

t (τ) solves (3.3.7).

Proposition 3.3.1. Let Assumption 3.3.1(β) hold for some β > 1 ∨ α. For any stoppingtime τ ≤ T and β′ ∈ [1 ∨ α, β), vt(x) = vt(τ, x) = Y−1

t (τ, x) solves (3.3.7) and for all ϵ > 0and p ≥ 2, there is a constant N = N(d1, p,N0,T, β′, η, ϵ) such that


1 vt(τ)|p0

]+ E

[supt≤T|r−ϵ1 ∇vt(τ)|pβ′−1

]≤ N. (3.3.9)

Proof. The estimate (3.3.9) is given in Theorem 2.1 in [LM14c], so we only need to showthat Y−1

t (τ, x) solves (3.3.7). Let (δn)n≥1 be a sequence such that δn ∈ (0, η) for all n andδn → 0 as n→ ∞. It is clear that there is a constant N = N(N0) such that for all ω and t,

π(z : Kt(z) > δn) ≤Nδαn. (3.3.10)


For each n, consider the system of SIDEs on [0,T ] × Rd1 given by

dv(n)t (x) =

(12(α)

12σ

iϱt (x)σ jϱ

t (x)∂i jv(n)t (x) + bit(x)∂iv

(n)t (x)

)dt

+1(1,2](α)∫

Z1Kt>δn(z)

(v(n)

t (x + Ht(x, z)) − v(n)t (x) + F i

t(x, z)∂iv(n)t (x)

)π(dz)dt

+

∫Z

1Kt>δn(z)(v(n)

t− (x + Ht(x, z)) − v(n)t− (x)

)[1(1,2](α)q(dt, dz) + 1[0,1](α)p(dt, dz)],

+12(α)σiϱt (x)∂iv

(n)t (x)dwϱ

t , τ < t ≤ T, v(n)t (x) = x, t ≤ τ, (3.3.11)

and the stochastic flow Y (n)t = Y (n)

t (τ, x), (t, x) ∈ [0,T ] × Rd1 , generated by the SDE

dY (n)t = −1[1,2](α)bt(Y

(n)t )dt − 12(α)σϱ

t (Y (n)t )dwϱ

t

+

∫Z

1Kt>δn(z)Ft(Y(n)t− , z)[1(1,2](α)q(dt, dz) + 1[0,1](α)p(dt, dz)], τ < t ≤ T,

Y (n)t (x) = x, t ≤ τ. (3.3.12)

Since (3.3.10) holds, we can rewrite equation (3.3.12) as

dY (n)t = −

(1[1,2](α)bt(Y

(n)t ) + 1(1,2](α)

∫Z

1Kt>δn(z)Ft(Y(n)t , z)π(dz)

)dt (3.3.13)

− 12(α)σϱt (Yn(t))dwϱ

t +

∫Z

1Kt>δn(z)Ft(Y(n)t− , z)p(dt, dz), τ < t ≤ T,

and (3.3.11) as

dv(n)t (x) =

(12(α)

12σ

iϱt (x)σ jϱ

t (x)∂i jv(n)t (x) + bit(x)∂ jσ

iϱt (x)

)dt

+ 12(α)σiϱt (x)∂iv

(n)t (x)dwϱ

t + 1(1,2](α)∫

Z1Kt>δn(z)F i

t(x, z)π(dz)∂iv(n)t (x)dt

+

∫Z

1Kt>δn(z)(v(n)

t− (x + Ht(x, z)) − v(n)t− (x)

)p(dt, dz), τ < t ≤ T. (3.3.14)

It is well-known that the solution Y (n)t = Y (n)

t (x) of (3.3.13) can be written as the solution ofcontinuous SDEs with a finite number of jumps interlaced. Indeed, for each n and stoppingtime τ′ ≤ T , consider the stochastic flow Y (n)

t = Y (n)t (τ′, x), (t, x) ∈ [0,T ] × Rd1 , generated

by the SDE

dY (n)t = −

(1[1,2](α)bt(Y

(n)t ) + 1(1,2](α)

∫Z

1K>δn(t, z)Ft(Y(n)t , z)π(dz)

)dt

− 12(α)σϱt (Y (n)

t )dwϱt , τ

′ < t ≤ T,

Y (n)t = x, t ≤ τ′.


By Theorems 2.1 and 2.4 and Remark 2.5, there is a modification of Y (n)t = Y (n)

t (τ′, x), stilldenoted Y (n)

t (τ′, x), that is a Cβ′

loc-diffeomorphism. Furthermore, P-a.s. we have that

Y (n)· (τ′, ·), Y (n);−1

· (τ′, ·) ∈ C([0,T ];Cβ′

loc)

and v(n)t = v(n)

t (τ′, x) = Y (n);−1t (τ′, x) solves the SPDE given by

dv(n)t (x) =

(12(α)

12σ

iϱt (x)σ jϱ

t (x)∂i jv(n)t (x) + bit(x)∂iv

(n)t (x)

)dt

+ 12(α)σiϱt (x)∂iv

(n)t (x)dwϱ

t

+ 1(1,2](α)∫

Z1K>δn(t, z)F i(t, z)π(dz)∂iv

(n)t (x)dt, τ′ < t ≤ T,

v(n)t (x) = x, t ≤ τ′.

For each n, let

A(n)t =

∫]0,t]

∫Z

1Ks>δn(z)p(ds, dz), t ≥ 0,

and define the sequence of stopping times (τ(n)l )∞l=1 recursively by τ(n)

0 = τ and

τ(n)l+1 = inf

t > τ(n)

l : ∆A(n)t , 0

∧ T.

Fix some n ≥ 1. It is clear that P-a.s. for all x and t ∈ [0, τ(n)1 ),

Y (n);−1t (τ, x) = Y (n);−1

t (τ, x) = v(n)t (τ, x)

satisfies (3.3.14) up to, but not including time τ(n)1 . Moreover, P-a.s. for all x,

Y (n)τ(n)

1

(τ, x) = Y (n)τn

1−(τ, x) +

∫Z

Fτ(n)1

(Y (n)τ(n)

1 −(τ, x), z)p(τ(n)

1 , dz),

and henceY (n);−1τ(n)

1

(τ, x) =∫

Zv(n)τ(n)

1 −(τ, x + Hτ(n)

1(x, z))p(τ(n)

1 , dz).

Consequently, v(n)t (τ, x) = Y (n);−1

t (τ, x) solves (3.3.14) up to and including time τ(n)1 . Assume

that for some l ≥ 1, v(n)t (τ, x) = Y (n);−1

t (τ, x) solves (3.3.14) up to and including time τ(n)l .

Clearly, P-a.s. for all x and t ∈ [τ(n)l , τ(n)

l+1), Y (n)t (x) = Y (n)

t (τ(n)l ,Y (n)

τ(n)l −

(x)), and thus P-a.s. for

all x and t ∈ [τ(n)l , τ(n)

l+1),

Y (n);−1t (x) = Y (n)

t (τ(n)l ,Y (n)

τ(n)l −

(x)) = v(n)t (τ(n)

l ,Y (n)τ(n)

l −(x)).


Moreover, P-a.s. for all x,

Y−1n (τn

l+1, x) =∫

Uvn(τn

l , τnl+1−, x + H(τn

l+1, x, z))p(τnl+1, dz),

which implies that v(n)t (τ, x) = Y (n);−1

t (τ, x) solves (3.3.14) up to and including time τnl+1.

Therefore, by induction, for each n, v(n)t (τ, x) = Y (n);−1

t (τ, x) solves (3.3.14). It is easy tosee that for all ω, t, and z,

|r−11 1Kt>δn(z)Ft(z) − r−1

1 Ft(z)|0 + |1Kt>δn(z)∇Ft(z) − ∇Ft(z)|β−1 ≤ 1Kt≤δn(z)Kt(z)

and thus

dPdt − limn→∞

∫D

1K≤δn(t, z)Kt(z)2π(dz) + dPdt − limn→∞

∫E

1K≤δn(t, z)Kt(z)π(dz) = 0.

By virtue of Theorem 2.3 in [LM14c], for all ϵ > 0 and p ≥ 2 we have

limn→∞

(E


1 (Y (n)t (τ) − r−(1+ϵ)

1 Yt(τ)|p0

]+ E

[supt≤T|r−ϵ1 ∇Y (n)

t (τ) − r−ϵ1 ∇Yt(τ)|pβ′−1

])= 0,

limn→∞


1 Y (n);−1t (τ) − r−(1+ϵ)

1 Y−1t (τ)|p0

]= 0

and

limn→∞

E[supt≤T|r−ϵ1 ∇Y (n);−1

t (τ) − r−ϵ1 ∇Y−1t (τ)|pβ′−1

]= 0.

Then passing to the limit in both sides of (3.3.11) and making use of Assumption 3.3.1(β),the estimate (3.3.4), and basic convergence properties of stochastic integrals, we discoverthat vt(τ, x) = X−1

t (τ, x) solves (3.3.7) .

3.3.3 Adding free and zero-order terms

Set (wϱ)ϱ∈N = (w1;ϱ)ϱ∈N, (Z,Z, π) = (Z1,Z1, π1), p(dt, dz) = p1(dt, dz), and q(dt, dz)= p1(dt, dz) − π1(dz)dt. Also, set D = D1, E = E1, and assume Z = D ∪ E. Let υt(x) =(υllϱ

t (ω, x))1≤l,l≤d2, ϱ∈N be a ℓ2(R2d2)-valued RT ⊗ B(Rd1)-measurable function defined onΩ×[0,T ]×Rd1 and ρt(x, z) = (ρll

t (ω, x, z))1≤l,l≤d2 be a PT ⊗B(Rd1)⊗Z-measurable functiondefined on Ω × [0,T ] × Rd1 × Z.

We introduce the following assumptions for β > 1 ∨ α and β > α.

Assumption 3.3.2 (β). (1) There is a constant N0 > 0 such that for all (ω, t) ∈ Ω× [0,T ],

|r−11 bt|0 + |r−1

1 σt|0 + |∇bt|β−1 + |∇σt|β−1 ≤ N0.



|r−11 Ht(z)|0 ≤ Kt(z) and |∇Ht(z)|β−1 ≤ Kt(z),


Kt(z) + Kt(z) +∫

D

(Kt(z)α + Kt(z)2

)π(dz) +

∫E

(Kt(z)α∧1 + Kt(z)

)π(dz) ≤ N0,

for all (ω, t, z) ∈ Ω × [0,T ] × Z.(2) There is a constant η ∈ (0, 1) such that for all (ω, t, x, z) ∈ (ω, t, x, z) ∈ Ω × [0,T ] ×

Rd1 × Z : |∇Ht(ω, x, z)| > η,

|(Id1 + ∇Ht(x, z)

)−1| ≤ N0.

Assumption 3.3.3 (β). There is a constant N0 > 0 such that for all (ω, t) ∈ Ω × [0,T ],

|ct|β + |υt|β + |r−θ1 ft|β + |r

−θ1 gt|β ≤ N0.


|ρt(z)|β + |r−θ1 ht(z)|β ≤ lt(z),

where l : Ω × [0,T ] × Z → R+ is a PT ⊗Z-measurable function satisfying

lt(z) +∫

Dlt(z)2π(dz) +

∫E

lt(z)π(dz) ≤ N0.

(ω, t, z) ∈ Ω × [0,T ] × Z.

Let Assumptions 3.3.2(β) and 3.3.3(β) hold for some β > 1 ∨ α and β > α. Let τ ≤ Tbe a stopping time and φ : Ω × Rd1 → Rd2 be a Fτ ⊗ B(Rd1)-measurable random field.Consider the system of SIDEs on [0,T ] × Rd1 given by

dvlt =

(Ll

tvt + bit∂iϕ

l + cllt ϕl + flt

)dt +

(N

lϱt vt + glϱ

t

)dwϱ

t

+

∫Z

(Il

t,zvt− + hlt(z)

)[1D(z)q(dt, dz) + 1E(z)p(dt, dz)], τ < t ≤ T,

vlt = φ

l, t ≤ τ, l ∈ 1, . . . , d2, (3.3.15)

where for ϕ ∈ C∞c (Rd1; Rd2) and l ∈ 1, . . . , d2,

Lltϕ(x) := 12(α)

12σ

iϱt (x)σ jϱ

t (x)∂i jϕl(x) + 12(α)σiϱ

t (x)allϱt (x)∂iϕ

l(x)


+

∫Dkρll

t (x, z)(ϕl(x + Ht(x, z)) − ϕl(x)

)π(dz)

+

∫Dk

(ϕl(x + Ht(x, z)) − ϕl(x) − 1(1,2](α)∂iϕ

l(x)Hit(x, z)

)π(dz)

Nlϱt ϕ

l(x) := 12(α)σiϱt (x)∂iϕ

l(x) + υllϱt (x)ϕl(x),

Ilt,zϕ

l(x) := (Id2 + ρllt (x, z))ϕl(x + Ht(x, z)) − ϕl(x),

and where

bit(x) : = 1[1,2](α)bi

t(x) + 12(α)σ jϱt (x)∂ jσ

iϱt (x)

+

∫D

(1(1,2](α)Hi

t(x, z) − Hit(H

−1t (x, z), z)

)π(dz),

cllt (x) : = cll

t (x) + 12(α)σ jϱt (x)∂ jυ

llϱt (x) +

∫D

(ρll

t (x, z) − ρllt (H−1

t (x, z), z))π(dz),

flt(x) : = f l

t (x) + 12(α)σ jϱt (x)∂ jgl

t(x) +∫

D

(hl

t(x, z) − hlt(H

−1t (x, z), z)

)π(dz).

We associate with (3.3.15) the stochastic flow Xt = Xt(x) = Xt (τ, x) , (t, x) ∈ [0,T ] × Rd1 ,given by (3.3.8). Let Γt(x) = Γt(τ, x), (t, x) ∈ [0,T ]×Rd1 , be the solution of the linear SDEgiven by

dΓt(x) = (ct(Xt(x))Γt(x) + ft(Xt(x))) dt +(υϱt (Xt(x))Γt(x) + gϱt (Xt(x))

)dwϱ

t

+

∫Zρt(H−1

t (Xt−(x), z), z)Γt−(x)[1D(z)q(dt, dz) + 1E(z)p(dt, dz)]

+

∫Z

ht(H−1t (Xt−(x), z), z)[1D(z)q(dt, dz) + 1E(z)p(dt, dz)], τ < t ≤ T,

Γt(x) = 0, t ≤ τ. (3.3.16)

Let Ψt(x) = Ψt(τ, x), (t, x) ∈ [0,T ] × Rd1 , be the unique solution of the linear SDE givenby

dΨt(x) = ct(Xt(x))Ψt(x)dt + υϱt (Xt(x))Ψt(x)dwϱt

+

∫Zρt(H−1

t (Xt−(x), z), z)Ψt−(x)[1D(z)q(dt, dz) + 1E(z)p(dt, dz)], τ < t ≤ T,

Ψt(x) = Id2 , t ≤ τ.

In the following lemma, we obtain p-th moment estimates of the weighted Höldernorms of Γ and Ψ.

Lemma 3.3.2. Let Assumptions 3.3.2(β) and 3.3.3(β) hold for some β > 1 ∨ α and β > α.For any stopping time τ ≤ T and β′ ∈ [0, β ∧ β), there exists a D([0,T ],Cβ

′

loc(Rd1; Rd2))-


modification of Γ(τ, ) and Ψ(τ), also denoted by Γ(τ) and Ψ(τ), respectively. Moreover, forall ϵ > 0 and p ≥ 2, there is a constant N = N(d1, d2, p,N0,T, β′, η, ϵ, θ) such that

E[supt≤T|r−(θ+ϵ)

1 Γt(τ)|pβ′

]+ E

[supt≤T|r−ϵ1 Ψt(τ)|pβ′

]≤ N. (3.3.17)

Proof. Let τ ≤ T be a fixed stopping time and β := β∧ β. Estimating (3.3.16) directly andusing the Burkholder-Davis-Gundy inequality and Lemma 3.4.1, we get

E[supt≤T|Γt(x)|p

]≤ NE

∫ (|Γt(x)|p + r1(Xt(x))pθ|r−θ1 ft|

p0

)dt + E

(∫r1(Xt(x))2θ|r−θ1 gt|

20dt

)p/2

+E(∫ ∫

D

∫r1(H−1

t (Xt(x)))2θ|r−θ1 ht|2∞π(dz)dt

)p/2

+E∫ ∫

D∪Er1(H−1

t (Xt(x)))pθ|r−θ1 ht|p∞π(dz)dt

+E(∫ ∫

Er1(H−1

t (Xt(x)))θ|r−θ1 ht|∞π(dz)dt)p

.

Then using multiplicative decomposition

ht(x, H−1t (Xt−(x), z), z) = r1(Xt−(x))θ

r1(H−1t (Xt−(x), z))θ

r1(Xt(x))θht(H−1

t (Xt−(x), z), z)r1(H−1

t (Xt−(x), z))θ,

Hölder’s inequality, Lemma 3.4.10 (1), Lemma 3.2 in [LM14c], and Gronwall’s inequality,we get that for all x and y,

E[supt≤T|Γt(x)|p

]≤ Nr−θp

1 (x),

where N = N(d1, p,N0,T, η, θ) is a positive constant. In a similar way, grouping terms inthe obvious manner and using Lemma 3.4.3 and Lemma 3.4.10 (3), we obtain

E[supt≤T|Γt(x) − Γt(y)|p

]≤ N(r−pθ

1 (x) ∨ r−pθ1 (y))|x − y|(β

′∧1)p.

Now, assume that [β]− ≥ 1. As in the proof of Theorem 3.4 in [Kun04], it follows thatUt = ∇Γt(τ, x) solves

dUt = (ct(Xt)Ut + ∇ct(Xt)∇XtΓt + ∇ ft(Xt)∇Xt) dt

+

∫Zρt(H−1

t (Xt−, z), z)Ut−[1D(z)q(dt, dz) + 1E(z)p(dt, dz)]

+

∫Z∇ρt(H−1

t (Xt−, z), z)∇[H−1t (Xt−)]Γt−[1D(z)q(dt, dz) + 1E(z)p(dt, dz)]

+

∫Z∇ht(x, H−1

t (Xt, z), z)∇[H−1t (Xt−)]][1D(z)q(dt, dz) + 1E(z)p(dt, dz)]


+

∫Z

ht(H−1t (Xt−(x), z), z)[1D(z)q(dt, dz) + 1E(z)p(dt, dz)], τ < t ≤ T,

Φt(x) = φ(x), t ≤ τ.

The following is a simple corollary of Lemma 3.3.2.

Corollary 3.3.3. Let Assumptions 3.3.2(β) and 3.3.3(β) hold for some β > 1∨α and β > α.For any stopping time τ ≤ T andFτ⊗B(Rd1)-measurable random field φ such that for someβ′ ∈ [0, β∧ β), P-a.s. φ ∈ Cβ

′

loc(Rd1; Rd2), there is a D([0,T ];Cβ

′

loc(Rd1 ,Rd2))-modification of

Φ(τ), also denoted by Φ(τ), and P-a.s. for all (t, x) ∈ [0,T ] × Rd1 ,

Φt(τ, x) = Ψt(x)φ(x) + Γt(x).

Moreover, if for some θ′ ≥ 0 and β′ ∈ [0, β ∧ β), P-a.s. r−θ′

1 φ ∈ Cβ′

(Rd1 ; Rd2), then for allϵ > 0 and p ≥ 2, there is a constant N = N(d1, d2, p,N0,T, θ, θ′, β′, ϵ) such that

E[supt≤T|r−(θ∨θ′)−ϵ

1 Φt(τ)|pβ′∣∣∣Fτ] ≤ N(|r−θ

′

1 φ|pβ′ + 1). (3.3.18)

Let us now state our main result concerning degenerate SIDEs and their connectionwith linear transformations of inverse flows of jump SDEs.

Proposition 3.3.4. Let Assumptions 3.3.2(β) and 3.3.3(β) hold for some β > 1 ∨ α andβ > α. For any stopping time τ ≤ T and Fτ ⊗ B(Rd1)-measurable random field φ suchthat for some β′ ∈ (α, β ∧ β) and θ′ ≥ 0, P-a.s. r−θ

′

1 φ ∈ Cβ′

(Rd1; Rd2), we have that P-a.s. Φ(τ, X−1(τ)) ∈ D([0,T ];Cβ

′

loc(Rd1; Rd2)) and vt(x) = vt(τ, x) = Φt(τ, X−1

t (τ, x)) solves(3.3.15). Moreover, for all ϵ > 0 and p ≥ 2,


1 vt(τ)|pβ′∣∣∣Fτ] ≤ N(|r−θ

′

1 φ|pβ′ + 1), (3.3.19)

for a constant N = N(d1, d2, p,N0,T, β′, η, ϵ, θ, θ′).

Proof. Fix a stopping time τ ≤ T and random field φ such that for some β′ ∈ (α, β ∧ β)and θ′ ≥ 0, P-a.s. r−θ

′

1 φ ∈ Cβ′

(Rd1; Rd2). By virtue of Corollary 3.3.3 and Theorem 2.1 in[LM14c], P-a.s.

Φ(τ, X−1(τ)) ∈ D([0,T ];Cβ′

loc(Rd1 ,Rd2)).

Then using the Ito-Wenzell formula (Proposition 3.4.16) and following a simple calcula-tion, we obtain that vt(τ, x) := Φt(τ, X−1

t (τ, x)) solves (3.3.15). By Theorem 2.1 in [LM14c]and Corollary 3.3.3, for all ϵ > 0 and p ≥ 2, there exists a constant N = N(d1, p,N0,T,


β′, η, ϵ) such that


1 X−1t (τ)|pβ′

]+ E


t (τ)|pβ′−1

]≤ N. (3.3.20)

Therefore applying Lemma 3.4.9 and Hölder’s inequalty and using the estimates (3.3.20)and (3.3.18), we procure (3.3.19), which completes the proof.

3.3.4 Adding uncorrelated part (Proof of Theorem 3.2.2)

Proof of Theorem 3.2.2 . Fix a stopping time τ ≤ T and random field φ such that for someβ′ ∈ (α, β∧ β) and θ′ ≥ 0, P-a.s. r−θ

′

1 φ ∈ Cβ′

(Rd1; Rd2). Consider the system of SIDEs givenby

dvlt =

((L1;l

t +L2;lt )vt + 1[1,2](α)bi

t∂iult + cll

t ult(x) + f l

t

)dt +

(N

1;lϱt vt + glϱ

t

)dw1;ϱ

t

+N2;lϱt vtdw2;ϱ

t +

∫Z1

(I

1;lt,z vt− + hl

t(z))

[1D1(z)q1(dt, dz) + 1E1 p1(dt, dz)]

+

∫Z2I

2;lt,z vt−[1D2(z)q2(dt, dz) + 1E2(z)p2(dt, dz)] τ < t ≤ T,

vlt = φ

l, t ≤ τ, l ∈ 1, . . . , d2,


N2;lϱt ϕ(x) := 12(α)σ2;iϱ


t (x)ϕl(x), ϱ ∈ N,

I2;lt,zϕ(x) := (Ill

d2+ ρ2;ll

t (x, z))ϕl(x + H2t (x, z)) − ϕl(x).

By Proposition 3.3.4, P-a.s. Φ(τ, X−1(τ)) ∈ D([0,T ];Cβ′

loc(Rd1; Rd2)) and vt(τ, x) = Φt(τ,

X−1t (τ, x)) solves (3.3.15). We write vt(x) = vt(τ, x). Moreover, for all ϵ > 0 and p ≥ 2,


1 vt(τ)|pβ′∣∣∣Fτ] ≤ N(|r−θ

′

1 φ|pβ′ + 1), (3.3.21)

where N = N(d1, d2, p,N0,T, β′, η1, η2, ϵ, θ, θ′) is a positive constant. Without loss of gen-erality we will assume that for all ω and t, |r−θ

′

1 φ|β′ ≤ N, since we can always multiply theequation by indicator function. For each n ∈ N0, let Cn

loc(Rd1; Rd2) be the separable Fréchet

space of n-times continuously differentiable functions f : Rd1 → Rd2 endowed with thecountable set of semi-norms given by

| f |n,k =∑

0≤|γ|≤n

sup|x|≤k|∂γ f (x)|, k ∈ N.


Owing to Lemma 3.4.2, there is a the family of measures Etω(dU), (ω, t) ∈ Ω × [0,T ]

on D([0,T ]; C[β]−

loc (Rd1; Rd2)), corresponding to A = v such that for all bounded G : Ω ×[0,T ]×[0,T ]×D([0,T ]; C[β]−

loc (Rd1; Rd2))→ Rd2 that are OT ×B ([0,T ])×B(D([0,T ]; C[β]−

loc

(Rd1; Rd2))) measurable, P-a.s. for all t, we have

Et[Gt(t, v)] =∫

D([0,T ];C[β′]−loc (Rd1 ;Rd2 ))

Gt(t,U)Et(dU) = E [Gt(t, v)|Ft] ,

where the right-hand-side is the càdlàg modification of the conditional expectation. Set

ut(x) = ut(τ, x) = Et[vt(τ, x)] =∫

D([0,T ];C[β′]−loc (Rd1 ;Rd2 ))

Ut(x)Et(dU).

Let λ = (θ ∨ θ′) + ϵ. We claim that for all multi-indices γ with |γ| ≤ [β]−, P-a.s. for all tand x,

∂γ[r−λ1 (x)ut(x)] =∫

D([0,T ];C[β′]−loc (Rd1 ;Rd2 ))

∂γ[r−λ1 (x)Ut(x)]Et(dU) = Et[∂γ[r−λ1 (x)vt(x)]].

Indeed, since

Mt = Et

[sups≤T|∂γ[r−λ1 vs]|0

], t ∈ [0,T ],

is a (F,P) martingale, we have

E[supt≤T|Mt|

2]≤ 4E

[|MT |

2]≤ 4E

[supt≤T|∂γ[r−λ1 vt]|20|

]< ∞, (3.3.22)

which implies that P-a.s. for all t,∫D([0,T ];C[β′]−

loc (Rd1 ;Rd2 ))sup

s≤T,x∈Rd1

|∂γ[r−λ1 (x)Us(x)]|Et(dU) = Et

[supt≤T|∂γ[r−λ1 vt]|0

]< ∞.

Similarly, since E[supt≤T |r

−λ1 vt|

2β′

]< ∞, P-a.s. for all x and y,

|∂γ[r−λ1 (x)ut(x)] − ∂γ[r−λ1 (y)ut(y)]||x − y|β′+

≤ Et

[|∂γ[r−λ1 (x)vt(x)] − ∂γ[r−λ1 (y)vt(y)]|

|x − y|β′+

]≤ Et[|r−λ1 vt|β′],

and thus, P-a.s.

supt≤T|r−λ1 ut|β′ ≤ sup

t≤TEt

[supt≤T|r−λ1 vt|β′

]< ∞.


Thus, P-a.s. r−λ1 (·)u(τ) ∈ D([0,T ];Cβ′

(Rd1; Rd2)) and (3.2.4) follows from (3.3.21) (see theargument (3.3.22)). For each l ∈ 1, . . . , d2, letAl

t(x) = Alt(z) be defined by

Alt = φ

l +

∫]τ,τ∨t]

((L1;l

s +L2;ls )us + 1[1,2](α)bi

s∂iuls + cll

s uls + f l

s

)ds

+

∫]τ,τ∨t]

(N1;lϱ

s us + glϱs

)dw1;ϱ

s

+

∫]τ,τ∨t]

∫Z1

(I1;l

s,zus− + hls(z)

)[1D1(z)q1(ds, dz) + 1E1(z)p1(ds, dz)].

By Theorem 12.21 in [Jac79], the representation property holds for (F,P), and hence everybounded (F,P)- martingale issuing from zero can be represented as

Mt =

∫]0,t]

oϱsdw1;ϱs +

∫]0,t]

∫Z1

es(z)q1(ds, dz), t ∈ [0,T ],

whereE

∫]0,T ]|os|

2ds + E∫

]0,T ]

∫Z1|es(z)|2π1(dz)ds < ∞.

Then for an arbitrary F-stopping time τ ≤ T and bounded (F,P)- martingale, applyingItô’s product rule and taking the expectation, we obtain

Evτ(τ, x)Mτ = EAτ(x)Mτ.

Since the optional projection is unique (see Theorem 13 in Chapter 1, Section 8 in [LS89]),P-a.s. for all t and x, ut(x) = At(x). This completes the proof.

3.3.5 Proof of Theorem 3.2.5

Proof of Theorem 3.2.5. Fix a stopping time τ ≤ T and random field φ such that for someβ′ ∈ (α, β ∧ β) and θ′ ≥ 0, P-a.s. r−θ

′

1 φ ∈ Cβ′

(Rd1; Rd2). For any δ > 0, we can rewrite(3.1.1) as

dult =

((L1;l

t +L2;lt )ut + 1[1,2](α)bi

t∂iult + cll

t ult + f l

t

)dt +

(N

1;lϱt ut + glϱ

t

)dw1;ϱ

t

+

∫Z1

(I

1;lt,z ut− + hl

t(z))

[1D1(z)q1(dt, dz) + 1E1(z)p1(dt, dz)]

+

∫Z1

(1(D1∪E1)∩K1

t >δ(z) + 1V1(z)

) (I

1;lt,z ut− + hl

t(z))

p1(dt, dz), τ < t ≤ T,

ult = φ

l, t ≤ τ, l ∈ 1, . . . , d2, (3.3.23)



L1;lt ϕ(x) := 12(α)

12σ

1;iϱt (x)σ1; jϱ

t (x)∂i jϕl(x) + 12(α)σk;iϱ

t (x)υ1;llϱt (x)∂iϕ

l(x)

+

∫D1ρ1;ll

t (x, z)(ϕl(x + H1

t (x, z)) − ϕl(x))π1(dz)

+

∫D1

(ϕl(x + H1

t (x, z)) − ϕl(x) − 1(1,2](α)H1;it (x, z)∂iϕ

l(x))π1(dz),

I1t,zϕ

l(x) = (Illd2+ 1K1

t ≤δ(z)ρ1;ll

t (x, z))ϕl(x + 1K1t ≤δ

(z)H1t (x, z)) − ϕl(x),

H1 := 1K1t ≤δ

H1, ρ1 := 1K1t ≤δ

ρ1, h := 1K1t ≤δ

h,

bit(x) := bi

t(x) −∫

D1∩K1t >δ

1(1,2](α)H1;it (x, z)π1(dz),

cllt (x) := cll

t (x) −∫

D1∩K1t >δ

ρ1;llt (x, z)π1(dz).

For an arbitrary stopping time τ′ ≤ T and Fτ′ ⊗ B(Rd1)-measurable random field φτ′

:Ω × Rd1 → Rd2 satisfying for some θ(τ′) > 0, P-a.s. r−θ(τ

′)1 φτ

′

∈ Cβ′

(Rd1; Rd2), consider thesystem of SIDEs on [0,T ] × Rd1 given by

dvlt =

((Lt

1;l+L2;l

t )vt + 1[1,2](α)bit∂ivl

t + cllt vl

t + f lt

)dt +

(N

1;lϱt vt + glϱ

t

)dw1;ϱ

t

+

∫Z1

(I

1;lt,z ut− + hl

t(z))

[1D1(z)q1(dt, dz) + 1E1(z)p1(dt, dz)], τ′ < t ≤ T,

vlt = φ

τ′;l, t ≤ τ′, l ∈ 1, . . . , d2. (3.3.24)

Set H2 = H2 and ρ2 = ρ2. In order to invoke Theorem 3.2.2 and obtain a unique solutionvt = vt(τ′, x) = vt(τ′, φτ

′

, x) of (3.3.24), we will show that for all ω and t,

|r−11 bt|0 + |∇bt|β−1 + |ct|β + |r

−θ f |β ≤ N0, (3.3.25)

where

bit(x) : = 1[1,2](α)bi

t(x) −2∑

k=1

12(α)σk; jϱt (x)∂ jσ

k;iϱt (x)

−

2∑k=1

∫Dk

(1(1,2](α)Hk;i

t (x, z) − Hk;it ( ˜Hk;−1


cllt (x) : = cll

t (x) −2∑

k=1

12(α)σk;iϱt (x)∂iυ

k;llϱt (x)


−

2∑k=1

∫Dk

(ρk;ll

t (x, z) − ρk;llt ( ˜Hk;−1


f lt (x) : = f l

t (x) − σ1; jϱt (x)∂ jgl

t(x) −∫

D1

(hl

t(x, z) − hlt( ˜H1;−1

t (x, z), z))π1(dz).

Owing to Assumption 3.2.3(β, δ1), we easily deduce that there is a constant N = N(d1, N0,

β) such that for each k ∈ 1, 2 and all ω and t,

|σk; jϱt ∂ jσ

k;ϱt |β + |σ

k; jϱt ∂ ja

k;ϱt (x)|β + |σ

1; jϱt ∂ jg

ϱt |β ≤ N, if α = 2.

Since |∇H1t |0 ≤ δ, for any fixed η1 < 1, for all (ω, t, x, z) ∈ (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 ×

(D1 ∪ E1) : |∇H1t (ω, x, z)| > η1,∣∣∣∣(Id1 + ∇H1

t (ω, x, z))−1∣∣∣∣ ≤ 1

1 − δ.

Appealing to Assumption 3.2.3(β, δ1) and applying Lemma 3.4.10, we obtain that there isa constant N = N(d1, d2,N0) such that for each k ∈ 1, 2 and all ω, t, and z,

|Hk;it (z) − Hk;i

t ( ˜Hk;−1t (z), z)|β ≤ N(Kk

t (z) + Kkt (z))2 + N1(0,1](β+ + δk)Kk

t (z)Kkt (z)δ

k

+ N1(1,2](β+ + δk)(Kk

t (z)Kkt (z)δ

k+ Kk

t (z)2),

|ρkt (z) − ρk

t ( ˜Hk;−1t (z), z)|β ≤ Nlk

t (z)(Kkt (z) + Kk

t (z)) + N1(0,1](β+ + µk)lkt (z)Kk

t (z)µk

+ N1(1,2](β+ + µk)(lkt (z)Kk

t (z)µk+ lk

t (z)Kkt (z)

),

and

|r−θ1 ht(z) − r−θ1 ht( ˜H1;−1t (z), z)|β ≤ Nl1

t (z)(K1t (z) + Kk

t (z)) + N1(0,1](β+ + µ1)lkt (z)K1

t (z)µ1

+ N1(1,2](β+ + µ1)(lkt (z)K1

t (z)µ1+ lk

t (z)K1t (z)

).

Moreover, using Lemma 3.4.10, we find that there is a constant N = N(d1, d2,N0) suchthat for each k ∈ 1, 2 and all ω, t, and z,

|r−11 Hk

t ( ˜Hk;−1t (z), z)|0 ≤ |r−1

1 Hk|0, |∇[Hk;it ( ˜Hk;−1

t (z), z)]|β ≤ |∇Hk|β−1.

Combining the above estimates and using Hölder’s inequality and the integrability prop-erties of lk

t (z) and Kkt (z), we obtain (3.3.25). Therefore, by Theorem 3.2.2, for each stop-

ping time τ′ ≤ T and and Fτ′ ⊗ B(Rd1)-measurable random field φτ′

satisfying for someθ(τ′) > 0, P-a.s. r−θ(τ

′)1 φτ

′

∈ Cβ′

(Rd1; Rd2), there exists a unique solution vt(x) = vt(τ′, φτ′

, x)

3.4. Appendix 66

of (3.3.24) such that

E[supt≤T|r−θ(τ

′)∨θ−ϵ1 vt(τ′)|

pβ′

∣∣∣Fτ′] ≤ N(|r−θ(τ′)

1 φτ′

|pβ′ + 1), (3.3.26)

where N = N(d1, d2, p,N0,T, β′, η1, η2, ϵ, θ, θ(τ′)) is a positive constant. Let

At =

∫]0,t]

∫Z1

(1(D1∪E1)∩K1

s>η1(z) + 1V1(z)

)p1(ds, dz), t ≤ T.

Define a sequence of stopping times (τn)n≥0 recursively by τ1 = τ and

τn+1 = inf(t > τn : ∆At , 0) ∧ T.

We construct the unique solution u = u(τ) of (3.3.23) in Cβ′

(Rd1; Rd2) by interlacing solu-tions of (3.3.24) along the sequence of stopping times (τn). For (ω, t) ∈ [[0, τ1)), we setut(τ, x) = vt(τ, φ, x) and note that

E[supt≤τ1

|r−θ′∨θ−ϵ

1 ut(τ)|pβ′∣∣∣Fτ] ≤ N(|r−θ

′

1 φ|pβ′ + 1).

For each ω and x, we set

uτ1(x) = uτ1−(x)

+

∫Z1

(1(D1∪E1)∩K1>η1(τ1, z) + 1V1(z)

) (I1τ1,zuτ1−(x) + hl

τ1(x, z)

)p1(τ1, dz).

By virtue of Lemma 3.4.9, there is a constant N = N(d1, d2, θ, θ′, ζτ1(z), β′)

|uτ1− H1τ1

(z) · r−ξτ1 (z)(θ∨θ′+ϵ+β′)1 |β′ ≤ N |r−θ∨θ

′−ϵ1 ul

τ1−|β′ ,

and hence|r−λ1

1 uτ1(x)|β′ ≤ N |r−θ∨θ′−ϵ

1 ulτ1−|β′ + ζτ1(z),

whereλ1 = (ξτ1(z)(θ ∨ θ′ + 1 + ϵ + β′)) ∨ θ ∨ (θ ∨ θ′ + ϵ).

We then proceed inductively, each time making use of the estimate (3.3.26), to obtain aunique solution u = u(τ) of (3.3.23), and hence (3.1.1), in Cβ

′

(Rd1; Rd2). This completesthe proof of Theorem 3.2.5.

3.4 Appendix

3.4. Appendix 67

3.4.1 Martingale and point measure moment estimates

Set (Z,Z, π) = (Z1,Z1, π1), p(dt, dz) = p1(dt, dz), and q(dt, dz) = q1(dt, dz). The follow-ing moment estimates are used to derive the estimates of Γt and Ψt in Lemma 3.3.2. Thenotation a ∼

p,Tb is used to indicate that the quantity a is bounded above and below by a

constant depending only on p and T times b.

Lemma 3.4.1. Let h : Ω × [0,T ] × Z → Rd1 be PT ⊗Z-measurable

(1) For any stopping time τ ≤ T and p ≥ 2,

E[supt≤τ

∣∣∣∣∣∣∫

]0,τ]

∫Z

hs(z)q(ds, dz)

∣∣∣∣∣∣p]∼p,T

E[∫

]0,τ]

∫Z|hs(z)|p π(dz)ds

]+ E

(∫]0,τ]

∫Z|hs(z)|2 π(dz)ds

)p/2 .x(2) For any stopping time τ ≤ T and p ≥ 1,

Esup

t≤τ

(∫]0,τ]

∫Z|hs(z)|p(ds, dz)

)p ∼p,T

E[∫

]0,τ]

∫Z|hs(z)| p π(dz)ds

]+ E

(∫]0,τ]

∫Z|hs(z)|π(dz)ds

) p ,Proof. We will only prove part (2), since part (1) is well-known (see, e.g., [Nov75] or[Kun04]). Assume that ht(ω, z) > 0 for all ω, t and z. Let

At =

∫]0,t]

∫Z

hs(z)p(ds, dz) and Lt =

∫]0,t]

∫Z

hs(z)π(dz)ds, t ≤ T.

It suffices to prove (2) for p > 1, since the case p = 1 is obvious. Fix an arbitrary stoppingtime τ ≤ T and p > 1. For all t, we have

Apt =

∑s≤t

[(As− + ∆As)p

− Aps−].

Thus, by the inequality

bp ≤ (a + b)p − ap ≤ p(a + b)p−1b ≤ p2p−1[ap−1b + bp], a, b ≥ 0,

we get

Apt ≥

∫]0,t]

∫Z

hs(z)p p(ds, dz)

and

Apt ≤ p2p−2

[∫]0,t]

∫h

hs(z)p p(ds, dz) +∫ t

0

∫Z

Ap−1s− hs(z)p(ds, dz)

].

3.4. Appendix 68

Since At is increasing, we obtain

E∫

]0,τ]

∫Z

hs(z)p p(ds, dz) ≤ EApτ ≤ p2p−2E

[∫]0,τ]

∫Z

hs(z)p p(ds, dz) + Ap−1τ Lτ

].

It is easy to see that

E[Lpτ ] = pE

∫]0,τ]

Lp−1s dLs = pE

∫]0,τ]

Lp−1s dAs ≤ pE[Lp−1

τ Aτ].

Applying Young’s inequality, for any ε > 0, we get

Ap−1τ Lτ ≤ εAp

τ +(p − 1)p−1

εp−1 pp Lpτ and Lp−1

τ Aτ ≤ εLpτ +

(p − 1)p−1

εp−1 pp Apτ .

Combining the estimates for any ε1 ∈ (0, 1p ), we have

εp−11 pp(1 − pε1)p(p − 1)p−1 ELp

τ

∨ E∫

]0,τ]

∫Z

hs(z)p p(ds, dz) ≤ E[Apτ ].

and for any ε2 ∈ (0, 1p2p−2 )

E[Apτ ] ≤

p2p−2

(1 − p2p−2ε2)E

∫]0,τ]

∫Z

hs(z)p p(ds, dz) +(p − 1)p−1

εp−12 pp

Lpτ

,which completes the proof.

3.4.2 Optional projection

The following lemma concerning the optional projection plays an integral role in Sec-tion 3.3.4 and the proof of Theorem 3.2.2. For more information on the Skorokhod J1-topology, we refer the reader to Chapter 6, Section 1 of [LS89]. Also, we refer the readerto Theorem 5.3 [Kal97] for the construction of regular conditional probability measureson Borel spaces.

Lemma 3.4.2. (cf. Theorem 1 in [Mey76]) Let X be a Polish space and D ([0,T ];X)be the space of X-valued càdlàg trajectories with the Skorokhod J1-topology. If A is arandom variable taking values in D ([0,T ];X), then there exists a family of B([0,T ])×F -measurable non-negative measures Et(dU), (ω, t) ∈ Ω × [0,T ], on D ([0,T ];X) and arandom-variable ζ satisfying P (ζ < T ) = 0 such that Et(D ([0,T ];X)) = 1 for t < ζ

and Et(D([0,T ];X)) = 0 for t ≥ ζ. In addition, Et is càdlàg in the topology of weakconvergence, Et = Et+ for all t ∈ [0,T ], and for each continuous and bounded functionalF on D ([0,T ];X) , the process Et (F) is the càdlàg version of E[F (A) |Ft]. If G : Ω ×

3.4. Appendix 69

[0,T ] × [0,T ] × D ([0,T ];X) → Rd2 is bounded and O × B ([0,T ]) × B (D ([0,T ];X))-measurable, then ∫

D([0,T ];X)Gt(ω, t,U)Et(dU) = Et(Gt)

is the optional projection of Gt(A) = Gt(ω, t,A). Furthermore, if G = Gt(ω, t,U) isbounded and P × B([0,T ]) × B(D([0,T ];X))-measurable, then Et−(Gt) is the predictableprojection of Gt(A) = Gt(ω, t,A).

Proof. We follow the proof of Theorem 1 in [Mey76]. Since D([0,T ];X) is a Polish space,for each t ∈ [0,T ], there is family of probability measures Et

ω(dw), ω ∈ Ω, on D([0,T ];X)such that for each A ∈ B(D([0,T ];X)), Et(A) is Ft-measurable and P-a.s.

P (A ∈ A|Ft) = Et (A) .

For each ω ∈ Ω, let I (ω) be the set of all t ∈ (0,T ] such that for any bounded continuousfunction F on D(([0,T ];X), the function

r 7→ Erω(F) =

∫D([0,T ];X)

F(w)Er(dw)

has a right-hand limit on [0, s) ∩ Q and a left-hand limit on (0, s] ∩ Q for every rationals ∈ [0,T ] ∩ Q. Let ζ (ω) = sup (t : t ∈ I(ω)) ∧ T. It is easy to see that P (ξ < T ) = 0.We set Et

ω = 0 if ξ(ω) < t ≤ T . The function Etω has left-hand and right-hand limits for

all t ∈ Q ∩ [0,T ]. We define Etω = Et+

ω for each t ∈ [0,T ) (the limit is taken along therationals), and ET

ω is the left-hand limit at T along the rationals. The statement follows byrepeating the proof of Theorem 1 in [Mey76] in an obvious way.

3.4.3 Estimates of Hölder continuous functions

In the coming lemmas, we establish some properties of weighted Hölder spaces that areused Section 3.3.5 and the proof of Theorem 3.2.5.

Lemma 3.4.3. Let β ∈ (0, 1] and θ1, θ2 ∈ R with θ1 − θ2 ≤ β.

(1) There is a constant c1 = c1 (θ2, β) such that for all ϕ : Rd1 → R with |r−θ11 ϕ|0 +

[r−θ21 ϕ]β =: N1 < ∞,

|ϕ(x) − ϕ(y)| ≤ c1N1(r1(x)θ2 ∨ r1(y)θ2)|x − y|β,

for all x, y ∈ Rd1 .

3.4. Appendix 70

(2) Conversely, if ϕ : Rd1 → R satisfies |r−θ11 ϕ|0 < ∞ and there is a constant N2 such that

for all x, y ∈ Rd1 ,

|ϕ(x) − ϕ(y)| ≤ N2(r1(x)θ2 ∨ r1(y)θ2)|x − y|β,

then[r−θ2

1 ϕ]β ≤ c1|r−θ11 ϕ|0 + N2.

Proof. (1) For all x, y with r1 (x)θ2 ≥ r1 (y)θ2 , we have

|ϕ(x) − ϕ(y)| ≤ r1(x)θ2[r−θ21 ϕ]β|x − y|β + r1(y)θ1−θ2 |r−θ1

1 ϕ|0|rθ21 (x) − r1(y)θ2 |

≤ ([r−θ21 ϕ]β + c1|r−θ1 ϕ|0)r1(x)θ2 |x − y|β,

where c1 := 1 + supt∈(0,1)1−tθ2(1−t)β if θ2 ≥ 0 and c1 := 1 + supt∈(1,∞)

(tθ2−1)tβ

(t−1)β if θ2 < 0, whichproves the first claim. (2) For all x and y with r1(x)θ2 > r1(y)θ2 , we have

|r1(x)−θ2ϕ(x) − r1(y)−θ2ϕ(y)|

≤ r1(x)−θ2 |ϕ(x) − ϕ(y)| + r1(y)θ1−θ2 |r−θ11 (y)ϕ(y)||r1(y)θ2r1(x)−θ2 − 1|

≤ (c1|r−θ1ϕ|0 + N2)|x − y|β,

which proves the second claim.

Lemma 3.4.4. Let β, µ ∈ (0, 1] and θ1, θ2, θ3, θ4 ∈ R with θ1 − θ2 ≤ β, θ3 − θ4 ≤ µ, andθ3 ≥ 0. If ϕ : Rd1 → R and H : Rd1 → Rd1 are such that

|r−θ11 ϕ|0 + [r−θ2

1 ϕ]β =: N1 < ∞ and |r−θ31 H|0 + [r−θ4

1 H]µ =: N2 < ∞,

then|ϕ H · r−θ1θ3

1 |0 ≤ |r−θ11 ϕ|0(1 + |r−θ3

1 H|0) ≤ N1 (1 + N2)θ1

and there is a constant N = N(β, µ, θ1, θ2) such that

[ϕ H · r−θ2θ3−βθ41 ]βµ ≤ NN1(1 + N2)θ2+β.

Proof. For all x, we have

r1(H(x)) ≤ (1 + |r−θ31 H|0)r1(x)θ3 ≤ (1 + N2)r1(x)θ3 ,

and hence|ϕ H · r−θ1θ3

1 |0 ≤ |r−θ11 ϕ|0|r

θ11 H · r−θ1θ3

1 |0 ≤ N1(1 + N2)θ1 .

3.4. Appendix 71

Using Lemma 3.4.3, for all x and y, we get

|ϕ(H(x)) − ϕ(H(y))| ≤ NN1(r1(H(x)) ∨ r1(H(y)))θ2 |H(x) − H(y)|β

≤ NN1(1 + N2)θ2(r1(x) ∨ r1(y))θ2θ3 Nβ2 (r1(x) ∨ r1(y))βθ4 |x − y|βµ

≤ NN1(1 + N2)θ2+β(r1(x) ∨ r1(y))θ2θ3+βθ4 |x − y|βµ,

for some constant N = N(β, µ, θ1, θ2). Noting that

θ1θ3 − θ2θ3 − βθ4 = (θ1 − θ2)θ3 − βθ4 ≤ β(θ3 − θ4) ≤ βµ,

we apply Lemma 3.4.3 to complete the proof.

Remark 3.4.5. Let β ∈ (0, 1] and θ1, θ2 ∈ R. Then there is a constant N = N(β, θ1, θ2) suchthat for all ϕ : Rd1 → R with |r−θ1

1 ϕ|0 + [r−θ21 ϕ]β =: N1 < ∞, we have |r−θϕ|β ≤ NN1, where

θ = max θ1, θ2 . In particular, if in Lemma 3.4.4, θ1 = θ2 and θ4 ≥ 0, then

|ϕ H · r−θ1θ3−βθ4 |βµ ≤ NN1(1 + N2)θ1+β.

Proof. If θ2 ≥ θ1, then the claim is obvious and if θ1 > θ2, for all x and y, we find

|r1(x)−θ1ϕ(x) − r1(y)−θ1ϕ(y)| ≤ r1(x)θ2−θ1 |r1(x)−θ2ϕ(x) − r1(y)−θ2ϕ(y)|

+

∣∣∣∣∣∣ r(y)θ1−θ2

r(x)θ1−θ2− 1

∣∣∣∣∣∣ |r−θ11 ϕ|0 ≤ N1(1 + c1)|x − y|β,

where c1 := supt∈(0,1)1−tθ1−θ2(1−t)β

.

Lemma 3.4.6. For any θ ≥ 0 and β > 1 , there are constants N1 = N1(d1, θ, β) andN2(d1, θ, β) such that for all ϕ : Rd1 → R,

N1|r−θ1 ϕ|β ≤∑|γ|≤[β]−

|r−θ1 ∂γϕ|0 +

∑|γ|=[β]−

|r−θ1 ∂γϕ|β+ ≤ N2|r−θ1 ϕ|β. (3.4.1)

Proof. For any multi-index γ with |γ| ≤ [β]− and x, we have

∂γ(r−θ1 ϕ)(x) =∑

γ1+γ2+=γ|γ1 |≥1

r1(x)θ∂γ1(r−θ1 )(x)r1(x)−θ∂γ2ϕ(x) + r1(x)−θ∂γϕ(x).

It is easy to show by induction that for all multi-indices γ, |rθ1∂γ(r−θ1 )|1 < ∞. Moreover, for

all multi-indices γ with |γ| < [β]−,

|r−θ1 ∂γϕ|1 ≤ |∇(r−θ1 ∂

γϕ)| ≤ |r−θ1 ∇(r−θ1 )|0|r−θ1 ∂γ∇ϕ|0.

3.4. Appendix 72

Thus, for any multi-index γ with |γ| ≤ [β]−,

|∂γ(r−θ1 ϕ)|0 ≤∑

γ1+γ2+=γ|γ1 |≥1

|rθ1∂γ1(r−θ1 )|0|r−θ1 ∂

γ2ϕ|0 + |r−θ1 ∂γϕ|0

and for any multi-index γ with |γ| = [β]−,

|∂γ(r−θ1 ϕ)|β+ ≤∑

γ1+γ2+=γ|γ1 |≥1

|rθ1∂γ1(r−θ1 )|1|r−θ1 ∇(r−θ1 )|0|r−θ1 ∂

γ2∇ϕ|0 + |r−θ1 ∂γϕ|0.

This proves the leftmost inequality in (3.4.1). For all i ∈ 1, . . . , d and x,

r−θ1 ∂iϕ(x) = ∂i(r−θ1 ϕ)(x) − r1(x)−θϕ(x)r1(x)θ∂i(r−θ1 )(x).

It follows by induction that for all multi-indices γ with |γ| ≤ [β]− and x, r−θ1 ∂γϕ(x) is

a sum of ∂γ(r−θ1 ϕ)(x), a finite sum of terms, each of which is a product of one term ofthe form ∂γ(r−θ1 ϕ)(x), |γ| < |γ|, and a finite number of terms of the form ∂γ1(rθ1)∂γ2(r−θ1 ),|γ1|, |γ2| ≤ |γ|. Since for all multi-indices γ1 and γ2, we have |∂γ1(rθ1)∂γ2(r−θ1 )|1 < ∞, therightmost inequality in (3.4.1) follows.

Corollary 3.4.7. For any θ ≥ 0 and β > 1 , there are constants N1 = N1(d1, θ, β) andN2(d1, θ, β) such that for all ϕ : Rd1 → R,

N1|r−θ1 ϕ|β ≤ |r−θ1 ϕ|0 +

∑|γ|=[β]−

|r−θ1 ∂γϕ|β+ ≤ N2|r−θ1 ϕ|β.

Proof. It is well known that for an arbitrary unit ball B ⊂ Rd1 and any 1 ≤ k < [β]−, thereis a constant N such that for any ε > 0,

supx∈B,|γ|=k

|∂γϕ| ≤ N(ε supx∈B,|γ|=[β]−

|∂γϕ(x)| + ε−k supx∈B|ϕ(x)|).

Let U0 = x ∈ Rd1 : |x| ≤ 1 and U j = x ∈ Rd1 : 2 j−1 ≤ |x| ≤ 2 j, j ≥ 1. For all j, we have

supx∈U j,|γ|=k

|∂γϕ(x)| = supB⊆U j

supx∈B,|γ|=k

|∂γϕ(x)|

≤ N(ε supB⊆U j

supx∈B,|γ|=[β]−

|∂γϕ(x)| + ε−k supB⊆U j

supx∈B|ϕ(x)|)

≤ N(ε supx∈U j,|γ|=[β]−

|∂γϕ(x)| + ε−k supx∈U j

|ϕ(x)|).

3.4. Appendix 73

Since for every j,

2−θ/22− jθ supx∈U j,|γ|=k

|∂γϕ(x)| ≤ supx∈U j,|γ|=k

|r−θ∂γϕ(x)| ≤ 2θ

2−( j−1)θ supx∈U j,|γ|=k

|∂γϕ(x)|,

we see that

2−θ/2 supj

2− jθ supx∈U j,|γ|=k

|∂γϕ(x)| ≤ supj

supx∈U j,|γ|=k

|r−θ∂γϕ(x)| = |r−θ∂γϕ|0

≤ 2θ supj

2− jθ supx∈U j,|γ|=k

|∂γϕ(x)|,

and the statement follows.

Remark 3.4.8. If ϕ : Rd1 → R is such that |r−θ1ϕ|0 + |r−θ2∇ϕ|0 < ∞ for θ1, θ2 ∈ R withθ1 − θ2 ≤ 1, then

[r−θ2ϕ]1 ≤ N(|r−θ1ϕ|0 + |r−θ2∇ϕ|0)

Proof. Indeed, for all x and y, we have

|ϕ(x) − ϕ(y)| ≤ |r−θ2∇ϕ|0

∫ 1

0rθ2(x + s(y − x))ds|y − x|

≤ |r−θ2∇ϕ|0(r(y)θ2 ∨ r(x)θ2)|y − x|,

and hence the claim follows from Lemma 3.4.3.

Lemma 3.4.9. Let n ∈ N, β, µ ∈ (0, 1], θ3, θ4 ≥ 0 be such that θ3 − θ4 ≤ 1. There is aconstant N = N(d1, θ1, θ3, θ4, n, β) such that for all ϕ : Rd1 → R with r−θ1

1 ϕ ∈ Cn+β(Rd1 ,Rd1)and H : Rd1 → Rd1 with

|r−θ31 H|0 + |r

−θ41 ∇H|n−1+µ =: N2 < ∞,

we have|ϕ H · r−θ1θ3 |0 ≤ |r

−θ11 ϕ|0(1 + |r−θ3

1 H|0)θ1

and|r−θ1θ3−θ4(n+µ∧β)

1 ∇(ϕ H)|n−1+µ∧β ≤ N|r−θ11 ϕ|n+β(1 + N2)θ1+µ∧β+n.

Proof. It follows immediately from Lemma 3.4.4 and Remark 3.4.8 that

|ϕ H · r−θ1θ3 |0 ≤ |r−θ11 ϕ|0(1 + |r−θ3

1 H|0)θ1 .

Using induction, we get that for all x and |γ| = n,

∂γ(ϕ(H(x))) = Iγ1(x) + Iγ2(x) + Iγ3 (x),

3.4. Appendix 74

where

Iγ1(x) =

d1∑i=1

∂iϕ(H(x))∂γHi(x)

Iγ2(x) is a finite sum of terms of the form

∂i1 · · · ∂i|γ|ϕ(H(x))∂γ1 Hi1 · · · ∂γ|γ|Hi|γ|

with i1, . . . , i|γ| ∈ 1, 2, . . . , d, |γ1| = · · · = |γ|γ|| = 1, and∑|γ|

k=1 γk = γ, if n ≥ 2 and zerootherwise, and where Iγ3(x) is a finite sum of terms of the form

∂i1 · · · ∂ikϕ(H(x))∂γ1 Hi1(x) · · · ∂γk Hik(x)

with 2 ≤ k < n, i1, i2, . . . , ik ∈ 1, . . . , d, and∑k

j=1 γ j = γ, 1 ≤ |γ j| < |γ|, if n ≥ 3, and zerootherwise. Thus, owing to Lemmas 3.4.4 and 3.4.6, for any multi-index γ with |γ| = n, wehave

|r−θ3θ1−θ41 I

γ1 |0 ≤ N |r−θ1

1 ∇ϕ|0(1 + |r−θ31 H|0)θ1 |r−θ4

1 ∂γH|0,

|r−θ3θ1−nθ41 I

γ2 |0 ≤ N|r−θ1

1 ∂γϕ|0(1 + |r−θ31 H|0)θ1 |r−θ4

1 ∇H|n0,

and|r−θ3θ1−(n−1)θ4

1 Iγ3 |0 ≤ N|r−θ1

1 ϕ|n−1(1 + |r−θ31 H|0 + |r

−θ41 ∇H|)θ1+n−1,

and hence

|r−θ1θ3−nθ4∂γ(ϕ H)|0 ≤ N|r−θ11 ϕ|n(1 + |r−θ3

1 H|0 + |r−θ41 ∇H|)θ1+n.

Appealing again to Lemmas 3.4.4 and 3.4.6, for all multi-indices γ with |γ| = n, we get

|r−θ1θ3−(1+µ∧β)θ41 I

γ1 |µ∧β ≤ N|r−θ1

1 ϕ|1+µ∧β (1 + N2)θ1+µ∧β+1 ,

|r−θ1θ3−(n+µ∧β)θ41 I

γ2 |µ∧β + |r

−θ1θ3−(n−1+µ∧β)θ41 I

γ3 |µ∧β ≤ N|r−θ1

1 ϕ|n+µ∧β (1 + N2)θ1+n+µ∧β .

Then applying Lemmas 3.4.4 and 3.4.6, we complete the proof.

We shall now provide some useful estimates of composite functions of diffeomor-phisms.

Lemma 3.4.10. Let H : Rd1 → Rd1 be continuously differentiable and assume that for allx ∈ Rd1 ,

|H(x)| ≤ L0 + L1|x| and |∇H(x)| ≤ L2.

Assume that for all x ∈ Rd1 , κ(x) = (Id1 + ∇H(x))−1 exists and |κ(x)| ≤ Nκ.

(1) Then the mapping H(x) := x+H(x) is a diffeomorphism with H−1(x) = x−H(H−1(x))

3.4. Appendix 75

=: x + F(x) and for all x ∈ Rd1 ,

|F(x)| ≤ L0 + L1L0Nκ + L1Nκ|x|, |∇F(x)| ≤ NκL2, |(Id1 + ∇F(x)

)−1| ≤ 1 + L2.

For all p ∈ R, there is a constant N = N(L0, L1,Nκ, p) such that for all x ∈ Rd1 ,

rp1 (H(x))rp

1 (x)+

rp1 (H−1(x))

rp1 (x)

≤ N, r−11 (x)|Hi(x) + Fk;i(x)| ≤ N[H]1|r−1

1 H|0.

Moreover, there is a constant N = N(L0, L1,Nκ, p) such that∣∣∣∣∣∣rp1 (H)rp

1

− 1 + 1(1,2](α)pHir−21 xi

∣∣∣∣∣∣α

+

∣∣∣∣∣∣rp1 (H−1)

rp1

− 1 − 1(1,2](α)pF ir−21 xi

∣∣∣∣∣∣α

≤ N(|r−11 H|[α]−+1

0 + [H][α]−+11 ).

(2) If for some β > 1, |∇H|β−1 ≤ L3, then there is a constant N = N(d1, β,Nκ, L3) such that

|∇F|β−1 ≤ N |∇H|β−1. (3.4.2)

(3) If for some β ≥ 1, |∇H|β−1 ≤ L3, then for all θ ≥ 0, there is a constant N = N(d1, β,Nκ,

L1, L3, θ) such that ∣∣∣∣∣∣rθ1 H−1

rθ1− 1

∣∣∣∣∣∣β

≤ N(|r−11 H|0 + |∇H|β−1).

(4) If |H|0 ≤ L4, and for some β > 0, |∇H|β∨1−1 ≤ L5 and ϕ : Rd1 → R is such thatfor some µ ∈ (0, 1] and θ ≥ 0, r−θ1 ϕ ∈ Cβ+µ(Rd1; R), then there is a constant N =N(d1, β, µ,Nκ, L4, L5, θ) such that

|r−θ1 (ϕ H−1 − ϕ)|β ≤ N |r−θ1 ϕ|β(|H|0 + |∇H|β∨1−1)

+N1(0,1](β+ + µ)∑|γ|=[β]−

[∂γ(r−θ1 ϕ)]β++µLµ4

+N1(1,2](β+ + µ)∑|γ|=[β]−

([∇∂γ(r−θ1 ϕ)]β++µ−1Lµ4 + |∇∂

γ(r−θ1 ϕ)|0|∇H|0).

Proof. (1) Since (Id1+∇H(x))−1 exists for all x, it follows from Theorem 0.2 in [DMGZ94]that the mapping H is a global diffeomorphism. For all x, we easily verify H−1(x) =x − H(H−1(x)) by substituting H(x) into the expression. Simple computations show thatfor all x, we have

|∇H(x)| ≤ 1 + L2, |∇H−1(x)| = |κ(H−1(x))| ≤ Nκ,

3.4. Appendix 76

|∇F(x)| = |∇H(H−1(x))∇H−1(x)| ≤ NκL2,

|(Id1 + ∇F(x))−1| = |∇H−1(x)−1| = |κ(H−1(x))−1| = |Id1 + ∇H(H−1(x))| ≤ 1 + L2.

For all x and y, we easily obtain

|H(x) − H(y)| ≤ (1 + L2)|x − y|, |H−1(x) − H−1(y)| ≤ Nκ|x − y|,

and hence

N−1κ |x − y| ≤ |H(x) − H(y)|, (1 + L2)−1|x − y| ≤ |H−1(x) − H−1(y)|. (3.4.3)

Making use of (3.4.3), for all x, we get

N−1κ |x| ≤ L0 + |H(x)|, |H−1(x)| ≤ NκL0 + Nκ|x|, |x| ≤ L0 + L1|H−1(x)|,

and thus|F(x)| ≤ L0 + L1NκL0 + L1Nκ|x|.

The rest of the estimates then follow easily from the above estimates and Taylor’s theorem.(2) Using the chain rule, for all x, we obtain

∇F(x) = −∇H(H−1(x))∇H−1(x) = −∇H(H−1(x))κ(H−1(x)), (3.4.4)

and hence |∇F|0 ≤ Nκ|∇H|0. For all x and y, we have

κ(H−1(y)) − κ(H−1(x)) = κ(y)[∇H(H−1(x)) − ∇H(H−1(y))]κ(x),

and thus since [H−1]1 ≤ (1 + NκL3) by part (1), we have for all δ ∈ (0, 1 ∧ β],

[κ(H−1)]δ ≤ N2κ (1 + NκL3)δ[∇H]δ.

It follows that there is a constant N = N(Nκ, L3) such that for all δ ∈ (0, 1 ∧ β],

|∇F|δ ≤ N |H|δ.

It is well-known that the inverse map I on the set of invertible d1×d1 matrices is infinitelydifferentiable and for each n, there exists a constant N = N(n, d1) such that for all invertiblematrices A, the n-th derivative of I evaluated at A, denoted I(n)(A), satisfies

|I(n)(A)| ≤ N|A−n−1| ≤ N|A−1|n+1.

3.4. Appendix 77

Using induction we find that for all multi-indices γ with |γ| ≤ [β]− and for all x, ∂γF(x) isa finite sum of terms, each of which is a finite product of

∂γH(H−1(x)), κ(H−1(x))n, I(n−1)(I + ∇H(H−1(x))), |γ| ≤ |γ|, n ∈ 1, . . . , |γ|.

Therefore, differentiating (3.4.4) and estimating directly we easily obtain (3.4.2).(3) For each x, we have

r1(H−1(x))θ

r1(x)θ− 1 = r1(x)−θ

∫ 1

0r1(Gs(x))θ−2Gs(x)∗F(x)ds

=

∫ 1

0

rθ−11 (Gs(x))r1(x)θ−1 K(Gs(x))∗dsr1(x)−1F(x),

where Gs(x) := x + sF(x), s ∈ [0, 1], and J(x) := r1(x)−1x. According to part (1) and (2),we have |r−1

1 F|0 ≤ N|r−11 H|0 and |∇F|β−1 ≤ N|∇H|β−1, and hence

|r−11 Gs|0 ≤ N(1 + |r−1

1 H|0), |∇Gs(x)|β−1 ≤ N(1 + |∇H|β−1).

and|J Gs|β ≤ N(1 + |r−1

1 H|0 + |∇H|β−1),

for some constant N independent of s. Moreover, using Lemma 3.4.9 we find

|rθ−11 Gs · r1−θ

1 |β ≤ N(1 + |r−1

1 H|0 + |∇H|β−1

)θ+β.

The statement then follows.(4) First, we will consider the case θ = 0. By part (1), we have that for all µ ∈ (0, (β+µ)∧1],

|ϕ H−1 − ϕ|0 ≤ [ϕ]µ|H H−1|µ0 ≤ [ϕ]µ|H|

µ0.

First, let us consider the case β ≤ 1. For each x, let J(x) = ϕ(H−1(x))−ϕ(x). For all x andy, it is clear that

|J(x) − J(y)| ≤ A(x, y) + B(x, y) +C(x, y),

whereA(x, y) := |J(x)|1[L4,∞)(|x − y|), B(x, y) := |J(y)|1[L4,∞)(|x − y|),

andC(x, y) := |J(x) − J(y)|1[0,L4)(|x − y|).

Moreover, owing to part (1), if β + µ ≤ 1, then for all x and y, we have

A(x, y) ≤ [ϕ]β+µLβ+µ4 1[L4,∞)(|x − y|) ≤ [ϕ]β+µLµ4 |x − y|β+

,

3.4. Appendix 78

B(x, y) ≤ [ϕ]β+µLµ4 |x − y|β,

and

C(x, y) ≤ [ϕ]β+µ|[H−1]β+µ1 |x − y|β+µ1[0,L4)(|x − y|) + [ϕ]β+µ|x − y|β+µ1[0,L4)(|x − y|)

≤ N[ϕ]β+µLµ4 |x − y|β

for some constant N = N(µ,Nκ, L4). Using the identity

J(x) − J(y)

= −

∫ 1

0

(∇ϕ

(x − θH(H−1(x))

)− ∇ϕ

(y − θH(H−1(y))

))H(H−1(x))dθ

−

∫ 1

0∇ϕ

(y − θH(H−1(y))

)(H(H−1(y)) − H(H−1(x))),

and part (1), if β + µ > 1, then there is a constant N = N(µ,Nκ, L4) such that for all x andy,

|J(x) − J(y)|1[L4,∞)(|x − y|) ≤ N([∇ϕ]β+µ−1|x − y|β+µ−1L4

+ |∇ϕ|0|x − y|[H]1)1[L4,∞)(|x − y|)

≤ N[∇ϕ]β+µ−1Lµ4 |x − y|β + N|∇ϕ|0|∇H|0|x − y|.

Moreover, since

J(x) − J(y)

=

∫ 1

0∇ϕ

(H−1(x + θ(y − x))

) (∇H−1(x + θ(y − x)) − Id1

)(x − y)dθ

+

∫ 1

0

(∇ϕ

(H−1 (x + θ(y − x))

)− ∇ϕ (x + θ(y − x))

)(x − y)dθ,

by part (1) and (3.4.2), if β + µ > 1, we attain that there is a constant N = N(µ,Nκ, L4)such that for all x and y,

|J(x) − J(y)|1[0,L4)(|x − y|) ≤ (|∇ϕ|0|∇H|0 + [∇ϕ]β+µ−1Lβ+µ−14 )|x − y|1[0,L4)(|x − y|)

≤ |∇ϕ|0|∇H|0|x − y| + [∇ϕ]β+µ−1Lµ4 |x − y|β.

Combining the above estimates, we get that for all β ≤ 1 and µ ∈ (0, 1], there is a constantN = N(µ,Nκ, L4) such that

[ϕ H−1 − ϕ]β ≤ N1[0,1](β+ µ)[ϕ]β+µLµ4 + N1(1,2](β+ µ)([∇ϕ]β+µ−1 + |∇ϕ|0|∇H|0

). (3.4.5)

3.4. Appendix 79

This proves the desired estimate for β ≤ 1 and θ = 0. We now consider the case β > 1.For β > 1, it is straightforward to prove by induction that for all multi-indices γ with1 ≤ |γ| ≤ [β]− and for all x,

∂γ(ϕ(H−1))(x) = Jγ1 (x) +Jγ

2 (x) +Jγ3 (x) +Jγ

4 (x),

whereJ

γ1 (x) := ∂γϕ(H−1(x)),

Jγ2 (x) = ∂γϕ(H−1)(∂1H−1;1)γ1 · · · (∂dH−1;d)γd − 1,

Jγ3 (x) is a finite sum of terms of the form

∂ j1 · · · ∂ jkϕ(H−1(x))∂γ1 H−1; j1(x) · · · ∂γk H−1; jk(x)

with 1 ≤ k < [β]−, j1, . . . , jk ∈ 1, . . . , d, and∑k

j=1 γ j = γ, and J4(x) is a finite sum ofterms of the form

∂ j1 . . . ∂ j[β]−ϕ(H−1(x))∂i1 H−1; j1(x) · · · ∂i[β]−

H−1; j[β]− (x)

with i1, j1, . . . , i[β]− , j[β]− ∈ 1, . . . , d and at least one pair ik , jk. Since for all x,

∇H−1(x) = I + ∇F(x)

and (3.4.2) holds, there is a constant N = N(d1, β) such that

∑1≤|γ|≤β

4∑i=2

|Jγi |0 +

∑|γ|=β

4∑i=2

|Jγi |β+ ≤ N |∇ϕ|β−1|∇F|β−1 ≤ N |∇ϕ|β−1|∇H|β−1.

If β > 2, then for all multi-indices γ with 1 ≤ |γ| < [β]−, we get

|Jγ1 − ∂

γϕ|0 = |∂γϕ H−1 − ∂γϕ|0 ≤ [∂γϕ]1|H|0.

It is easy to see that there is a constant N = N(L4,Nκ) such that for all γ with |γ| = [β]−

and all µ ∈ (0, (β+ + µ) ∧ 1],

|Jγ1 − ∂

γϕ|0 = |∂γϕ H−1 − ∂γϕ|0 ≤ [∂γϕ]µ|H|

µ0.

Moreover, appealing to the estimate (3.4.5) we obtain

[Jγ1 − ∂

γϕ]β+

≤ N1[0,1](β+ + µ)[∂γϕ]β++µLµ4 + N1(1,2](β+ + µ)([∇∂γϕ]β++µ−1 + |∇∂

γϕ|0|∇H|0).

3.4. Appendix 80

Let us now consider the case θ > 0. The following decomposition obviously holds for allx:

r1(x)−θϕ(H−1(x)) − r1(x)−θϕ(x) = ϕ(H−1) − ϕ(x) +(r1(H−1(x))θ

r1(x)θ− 1

)ϕ(H−1(x)),

where ϕ = r−θ1 ϕ ∈ Cβ(Rd1 ; Rd1). Thus, to complete the proof we require

|ϕ H−1|β ≤ N |ϕ|β and

∣∣∣∣∣∣rθ1 H−1

rθ1− 1

∣∣∣∣∣∣β

≤ N(|H|0 + |∇H|β∨1−1).

The latter inequality was proved in part (3) and the first inequality follows from part (2)and Lemma 3.4.9.

Remark 3.4.11. Let H : Rd1 → Rd1 be continuously differentiable and assume that for allx,

|∇H(x)| ≤ η < 1.

Then for all x ∈ Rd1 ,

|(Id1 + ∇H(x))−1| ≤ |Id1 +

∞∑k=1

(−1)k∇H(x)k| ≤1

1 − η.

3.4.4 Stochastic Fubini theorem

Let m = (mϱ)t≤T , ϱ ∈ N, be a sequence of F-adapted locally square integrable continuousmartingales issuing from zero such that P-a.s. for all t ∈ [0,T ], ⟨mϱ1 ,mϱ2⟩t = 0 for ϱ1 , ϱ2

and ⟨mϱ⟩t = Nt for ϱ ∈ N, where Nt is a PT -measurable continuous increasing processesissuing from zero. Let η(dt, dz) be a F-adapted integer-valued random measure on ([0,T ]×E,B([0,T ])⊗E), where (U,U) is a Blackwell space. We assume that η(dt, dz) is optional,PT -sigma-finite, and quasi-left continuous. Thus, there exists a unique (up to a P-null set)dual predictable projection (or compensator) ηp(dt, dz) of η(dt, dz) such that µ(ω, t×U) =0 for all ω and t. We refer the reader to Chapter II, Section 1, in [JS03] for any unexplainedconcepts relating to random measures.

Let (X,Σ, µ) be a sigma-finite measure space; that is, there is an increasing sequenceof Σ-measurable sets Xn, n ∈ N, such that X = ∪∞n=1Xn and µ(Xn) < ∞ for each n. Letf : Ω × [0,T ] × X → Rd2 be RT ⊗ Σ-measurable, g : Ω × [0,T ] × X → ℓ2(Rd2) beRT ⊗ Σ/B(ℓ2(Rd2))-measurable, and h : Ω × [0,T ] × X × U → Rd2 be PT ⊗ Σ ⊗ U-measurable. Moreover, assume that for all t ∈ [0,T ] and x ∈ X, P-a.s.∫

]0,T ]|gt(x)|2dNt +

∫]0,T ]

∫U|ht(x, z)|2ηp(dt, dz) < ∞.

3.4. Appendix 81

Let F = Ft(x) : Ω × [0,T ] × X → Rd2 be OT ⊗ B(X)-measurable and assume that fordPµ-almost all (t, x) ∈ [0,T ] × X,

Ft(x) =∫

]0,t]gϱs(x)dmϱ

s +

∫]0,t]

∫U

hs(x, z)η(dt, dz),

where η(dt, dz) = η(dt, dz) − ηp(dt, dz).The following version of the stochastic Fubini theorem is a straightforward extension

of Lemma 2.6 [Kry11] and Corollary 1 in [Mik83]. See also Proposition 3.1 in [Zho13],Theorem 2.2 in [Ver12], and Theorem 1.4.8 in [Roz90]. Indeed, to prove it for a boundedmeasure we can use a monotone class argument as in Theorem 64 in [Pro05]. To handlethe general setting with possibly infinite µ, we use assumptions (ii) and (iii) below andtake limits on the sets Xn using the Lenglart domination lemma (Theorem 1.4.5 in [LS89])and the following well-known inequalities:

E supt≤T

∣∣∣∣∣∣∫

]0,t]gϱsdmϱ

s

∣∣∣∣∣∣ ≤ NE(∫

]0,T ]|gt(x)|2dmϱ

t

)1/2

E supt≤T

∣∣∣∣∣∣∫

]0,t]

∫U

ht(x, z)η(dt, dz)

∣∣∣∣∣∣ ≤ NE(∫

]0,T ]

∫U|ht(x, z)|2ηp(dt, dz)

)1/2

,

where τ ≤ T is an arbitrary stopping time and N = N(T ) is a constant independent of gand h.

Proposition 3.4.12 (c.f. Corollary 1 in [Mik83] and Lemma 2.6 in [Kry11]). Assume that

(1) P-a.s. for all n ≥ 1,∫Xn

(∫]0,T ]|gt(x)|2dNt

)1/2

µ(dx) +∫

Xn

(∫]0,T ]

∫U1

|ht(x, z)|2ηp(dt, dz))1/2

µ(dx) < ∞;

(2) P-a.s. ∫]0,T ]

(∫X|gt(x)|µ(dx)

)2

dt +∫

]0,T ]

∫U

(∫X|ht(x, z)|µ(dx)

)2

ηp(dt, dz);

(3) P-a.s. for al t ∈ [0,T ], ∫X|Ft(x)|µ(dx) < ∞.

Then P-a.s. for all t ∈ [0,T ],∫X

Ft(x)µ(dx) =∫

]0,t]

∫X

gϱs(x)µ(dx)dmϱs +

∫]0,t]

∫U

∫X

hs(x, z)µ(dx)η(dr, dz).

We obtain the following corollary by applying Minkowski’s integral inequaility.

3.4. Appendix 82

Corollary 3.4.13. Assume that P-a.s.∫X

(∫]0,T ]|gt(x)|2dNt

)1/2

µ(dx) +∫

X

(∫]0,T ]

∫U1

|ht(x, z)|2ηp(dt, dz))1/2

µ(dx) < ∞. (3.4.6)

Then P-a.s. for all t ∈ [0,T ],∫X

Ft(x)µ(dx) =∫

]0,t]

∫X

gϱs(x)µ(dx)dmϱs +

∫]0,t]

∫U

∫X

hs(x, z)µ(dx)η(dr, dz).

Remark 3.4.14. If µ is a finite-measure and P-a.s.∫X

∫]0,T ]|gt(x)|2dNtµ(dx) +

∫X

∫]0,T ]

∫U1

|ht(x, z)|2ηp(dt, dz)µ(dx) < ∞,

then (3.4.6) holds by Hölder’s inequality.

3.4.5 Itô-Wentzell formula

Definition 3.4.15. We say that an Rd1-valued F-adapted quasi-left continuous semimartin-gale Lt = (Lk

t )1≤k≤d1 , t ≥ 0, is of α-order for α ∈ (0, 2], if P-a.s. for all t ≥ 0,∑s≤t

|∆Ls|α < ∞

and

Lt = L0 +

∫]0,t]

∫Rd1

0

zpL(ds, dz), if α ∈ (0, 1),

Lt = L0 + At +

∫]0,t]

∫|z|≤1

zqL(ds, dz) +∫

]0,t]

∫|z|>1

zpL(ds, dz), if α ∈ [1, 2),

Lt = L0 + At + Lct +

∫]0,t]

∫|z|≤1

zqL(ds, dz) +∫

]0,t]

∫|z|>1

zpL(ds, dz), if α = 2,

where pL(dt, dz) is the jump measure of L with dual predictable projection πL(dt, dz), qL

(dt, dz) = pL(dt, dz) − πL(dt, dz) is a martingale measure, At = (Ait)1≤i≤d1 is a continuous

process of finite variation with A0 = 0, and Lct = (Lc;i

t )1≤i≤d1 is a continuous local martingaleissuing from zero.

Set (wϱ)ϱ∈N = (w1;ϱ)ϱ∈N, (Z,Z, π) = (Z1,Z1, π1), p(dt, dz) = p1(dt, dz), and q(dt, dz)= q1(dt, dz). Also, set D = D1, E = E1, and assume Z = D ∪ E.

Let f : Ω × [0,T ] × Rd1 → Rd2 be RT ⊗ B(Rd1)-measurable, g : Ω × [0,T ] × Rd1 →

ℓ2(Rd2) be RT ⊗ B(Rd1)/B(ℓ2(Rd2))-measurable, and h : Ω × [0,T ] × Rd1 × Z → Rd2 be

3.4. Appendix 83

PT ⊗ B(Rd1) ⊗Z-measurable. Moreover, assume that, P-a.s. for all x ∈ Rd1 ,∫]0,T ]| ft(x)|dt +

∫]0,T ]|gt(x)|2dt < ∞

+

∫]0,T ]

∫D|ht(x, z)|2π(dz)dt +

∫]0,T ]

∫E|ht(x, z)|π(dz)dt < ∞.

Let F = Ft(x) : Ω × [0,T ] × Rd1 → Rd2 be OT ⊗ B(Rd1)-measurable and assume that forall x, P-a.s. for all t,

Ft(x) = F0(x) +∫

]0,t]fs(x)ds +

∫]0,t]

gϱs(x)dwϱs

+

∫]0,t]

∫Z

hs(x, z)[1D(z)q(ds, dz) + 1E(z)p(ds, dz)].

For each n ∈ 1, 2, let Cnloc(R

d1; Rd2) be space of n-times continuously differentiable func-tions f : Rd1 → Rd2 . We now state our version of the Itô-Wentzell formula. For each ω, tand x, we denote ∆F(x) = Ft(x) − Ft−(x).

Proposition 3.4.16 (cf. Proposition 1 in [Mik83] ). Let (Lt)t≥0 be an Rd1-valued quasi-leftcontinuous semimartingale of order α ∈ (0, 2]. Assume that:

(1) (a) P-a.s. F ∈ D([0,T ];Cαloc(Rd; Rm) if α is fractional and F ∈ D([0,T ]; Cα

loc(Rd; Rm)

if α = 1, 2 ;

(b) for dPdt-almost-all (ω, t) ∈ Ω × [0,T ], ft(x) and gt(x) = (giϱt (x))ϱ∈N ∈ ℓ2(Rd2) are

continuous in x and

dPdt − limy→x

[∫D|ht(y, z) − ht(x, z)|2π(dz) +

∫E|ht(y, z) − ht(x, z)|π(dz)

]= 0;

(c) for all ϱ ∈ N and i ∈ 1, . . . , d1 and for dPd|⟨Lc;i,wϱ⟩|t-almost-all (ω, t) ∈ Ω ×[0,T ], giϱ

t ∈ C1loc(R

d; R), if α = 2, ;(2) for all compact subsets K of Rd1 , P-a.s.∫

]0,T ]supx∈K

(| ft(x)| + |gt(x)|2 +

∫D|ht(x, z)|2π(dz) +

∫E|ht(x, z)|π(dz)

)dt < ∞,

∑ϱ∈N

∫]0,T ]

supx∈K|∇giϱ

t (x)|d|⟨Lc;i,wϱ⟩|t +∑t≤T

|∆Ft|α∧1;K |∆Lt|α∧1 < ∞.

Then P-a.s for all t ∈ [0,T ],

Ft(Lt) = F0(L0) +∫

]0,t]fs(Ls)ds +

∫]0,t]

gϱs(Ls)dwϱs

3.4. Appendix 84

+

∫]0,t]

∫Z

hs(Ls−, z)[1D(z)q(dr, dz) + 1E(z)p(dr, dz)]

+

∫]0,t]

∂iFs−(Ls−)[1[1,2](α)dAis + 12(α)dLc;i

s ]

+∑s≤t

(Fs−(Ls) − Fs−(Ls−) − 1[1,2](α)∇Fs−(Ls−)∆Ls

)+ 12(α)

12

∫]0,t]

∂i jFs(Ls)d⟨Lc;i, Lc; j⟩s (3.4.7)

+ 12(α)∫

]0,t]∂igϱs(Ls)d⟨wϱ, Lc;i⟩s +

∑s≤t

(∆Fs(Ls) − ∆Fs(Ls−)) .

Proof. Since both sides have identical jumps and we can always interlace a finite set ofjumps, we may assume that |∆Lt| ≤ 1 for all t ∈ [0,T ]; that is, it is enough to prove thestatement for Lt = Lt −

∑s≤t 1[1,∞)(|∆Ls|)∆Ls, t ∈ [0,T ]. It suffices to assume that for some

K and all ω, |L0| ≤ K. For each R > K, let

τR = inf

t ∈ [0,T ] : |A|t + |⟨Lc⟩|t +∑s≤t

|∆Ls|α + |Lt| > R

∧ T

and note that P-a.s. τR ↑ T as R tends to infinity. If instead of L, f , g, h, and F, we takeL·∧τR , f 1(0,τR], gϱ1(0,τR], h1(0,τR], F1(0,τR], then the assumptions of the proposition hold forthis new set of processes. Moreover, if we can prove (3.4.7) for this new set of processes,then by taking the limit as R tends to infinity, we obtain (3.4.7). Therefore, we may assumethat for some R > 0, P-a.s. for all t ∈ [0,T ],

|A|t + |⟨Lc⟩|t +∑s≤t

|∆Ls|α + |Lt| ≤ R. (3.4.8)

Let ϕ ∈ C∞c (Rd1 ,R) have support in the unit ball in Rd1 and satisfy∫

Rd1ϕ(x)dx =

1, ϕ(x) = ϕ(−x), and ϕ(x) ≥ 0, for all x ∈ Rd1 . For each ε ∈ (0, 1), let ϕε(x) =ε−dϕ (x/ε) , x ∈ Rd1 . By Itô’s formula, for all x ∈ Rd1 , P-a.s. for all t ∈ [0,T ],

Ft(x)ϕε(x − Lt) = F0(x)ϕε(x − L0) −∫

]0,t]Fs−(x)∂iϕε(x − Ls−)dLi

s

+

∫]0,t]

ϕε (x − Ls) fs(x)ds +∫

]0,t]ϕε (x − Ls) gϱs(x)dwϱ

s

+ 12(α)12

∫]0,t]

Fs(x)∂i jϕε(x − Ls)d⟨Lc;i, Lc; j⟩s

+ 12(α)∫

]0,t]gϱs(x)∂iϕε(x − Ls)d⟨wϱ, Lc;i⟩s

+

∫]0,t]

∫Zϕε (x − Ls−) hs(x, z)[1D(z)q(dr, dz) + 1E(z)p(dr, dz)]

3.4. Appendix 85

+∑s≤t

∆Fs(x) (ϕε(x − Ls) − ϕε (x − Ls−))

+∑s≤t

Fs−(x) (ϕε(x − Ls) − ϕε(x − Ls−) + ∂iϕε (x − Ls−)∆Ls) .

Appealing to assumption (2) and (3.4.8) (i.e. for the integrals against F), we integrateboth sides of the above in x, apply Corollary 3.4.13 (see, also, Remark 3.4.14) and thedeterministic Fubini theorem, and then integrate by parts to get that P-a.s. for all t ∈ [0,T ],

F(ε)t (Lt) = F(ε)

0 (L0) +∫

]0,t]∇F(ε)

s− (Ls−)[1[1,2](α)dAis + 12(α)dLc;i

s ] +∫

]0,t]f (ε)s (Ls)dr

+

∫]0,t]

g(ε)s (Ls)dwϱ

s +

∫]0,t]

∫Z

h(ε)s (Ls−, z)[1D(z)q(dr, dz) + 1E(z)p(dr, dz)]

+ 12(α)12

∫]0,t]

∂i jF(ε)s (Ls)d⟨Lc;i, Lc; j⟩s + 12(α)

∫]0,t]

∂ig(ε);ϱs (Ls)d⟨wϱ, Lc;i⟩s

+∑s≤t

(∆F(ε)

s (Ls) − ∆F(ε)s (Ls−)

)+

∑s≤t

(F(ε)

s− (Ls) − F(ε)s− (Ls−) − 1[1,2](α)∇F(ε)

s− (Ls−)∆Ls

)(3.4.9)

where for all ω, t, x, and z,

F(ε)t (x) := ϕε ∗ Ft(x), f (ε)

t = ϕε ∗ ft(x), g(ε);ϱt (x) = ϕε ∗ gϱt (x), h(ε)

t (x, z) = ϕε ∗ ht(x, z),

and ∗ denotes the convolution operator on Rd1 . Let BR+1 = x ∈ Rd1 : |x| ≤ R+1. Owing toassumption (1)(a) and standard properties of mollifiers, for any multi-index γ with |γ| ≤ α,P-a.s. for all t,

|∂γF(ε)t (Lt)| ≤ sup

t≤Tsup

x∈BR+1

|∂γFt(x)| < ∞

and for all x,dPdt − lim

ε↓0|∂γF(ε)

t (x) − ∂γF(ε)t (x)| = 0.

Similarly, by assumption 1(b), dPdt-almost-all (ω, t) ∈ Ω × [0,T ],

| f (ε)t (Lt)| ≤ sup

x∈BR+1

| ft(x)| < ∞, |g(ε)t (Lt)| ≤ sup

x∈BR+1

|gt(x)| < ∞,∫D|hεt (Lt, z)|2π(dz) ≤ sup

x∈BR+1

∫D|ht(x, z)|2π(dz),∫

E|hεt (Lt, z)|π(dz) ≤ sup

x∈BR+1

∫E|ht(x, z)|π(dz)

3.4. Appendix 86

and for all x,

dPdt − limε↓0| f (ε)

t (x) − ft(x)| = 0, dPdt − limε→0|g(ε)

t (x) − gt(x)| = 0

and

dPdt − limε↓0

∫Z[1D(z)|h(ε)

t (x, z) − ht(x, z)|2 + 1E(z)|h(ε)t (x, z) − ht(x, z)|]π(dz) = 0,

where in the last-line we have also used Minkowski’s integral inequality and a standardmollifying convergence argument. Using assumption 1(d), for all ϱ ∈ N and i ∈ 1, . . . , d1

and for dPd|⟨Lc;i,wϱ⟩|t-almost-all (ω, t) ∈ Ω × [0,T ]

|∇g(ε);iϱt (Lt)| ≤ sup

x∈BR+1

|∇giϱt (x)|

and for all x,

dPd|⟨Lc;i,wϱ⟩|t − limε→0|∇g(ε);iϱ

t (x) − ∇giϱt (x)| = 0, if α = 2.

Owing to assumption 1(a) and (3.4.8), P-a.s.∑s≤t

|F(ε)s− (Ls) − F(ε)

s− (Ls−) − 1[1,2](α)∇F(ε)s− (Ls−)∆Ls|

≤ supt≤T|Ft|α;BR+1

∑s≤t

|∆Ls|α ≤ R sup

t≤T|Ft|α;BR+1 .

Since P-a.s. F ∈ D([0,T ];Cα(Rd; Rm), it follows that for all x, P-a.s. for all t,

limε↓0|∆Fε

t (x) − ∆Ft(x)| = 0.

By assumption (2), P-a.s for all t, we have∑s≤t

(∆F(ε)

s (Ls− + ∆Ls) − ∆F(ε)s (Ls−)

)≤

∑s≤t

|∆Ft|α∧1;BR+1 |∆Ls|α∧1.

Combining the above and using assumptions (1)(a) and (2) and the bounds given in (3.4.8)and the deterministic and stochastic dominated convergence theorem, we obtain conver-gence of all the terms in (3.4.9), which complete the proof.

Chapter 4The L2-Sobolev theory for parabolicSIDEs

4.1 Introduction

Let (Ω,F ,P) be a probability space with the filtration F = (Ft)0≤t≤T of sigma-algebrassatisfying usual conditions. In a triple of Hilbert spaces (Hα+µ,Hα,Hα−µ) with parametersµ ∈ (0, 1] and α ≥ µ, we consider a linear stochastic evolution equation given by

dut = (Ltut + ft) dVt + (Mtut− + gt) dMt, t ≤ T, (4.1.1)

u0 = φ,

where Vt is a continuous non-decreasing process, Mt is a cylindrical square integrable mar-tingale, L andM are linear adapted operators, and ϕ, f , and g are adapted input functions.

By virtue of Theorems 2.9 and 2.10 in [Gyö82], under some suitable conditions onthe data φ, f and g, if L satisfies a growth assumption and L andM satisfy a coercivitycondition in the triple (Hα+µ,Hα,Hα−µ), then there exists a unique solution (ut)t≤T of (4.1.1)that is strongly càdlàg in Hα and belongs to L2(Ω × [0,T ],OT , dVtdP; Hα+µ), where OT isthe optional sigma-algebra on Ω × [0,T ]. In this chapter, under a weaker assumption thancoercivity (see Assumption 4.2.1(α, µ)) and using the method of vanishing viscosity, weprove that there exists a unique solution (ut)t≤T of (4.1.1) that is strongly càdlàg in Hα′ forall α′ < α and belongs to L2(Ω × [0,T ], dVtdP; Hα). Furthermore, under some additionalassumptions on the operators L andM we can show that the solution u is weakly càdlàgin Hα.

The variational theory of deterministic degenerate linear elliptic and parabolic PDEswas established by O.A. Oleinik and E.V. Radkevich in [Ole65] and [OR71]. In [Par75],É. Pardoux developed the variational theory of monotone stochastic evolution equations,which was extended in [KR77, KR79, GK81, Gyö82] by N.V. Krylov, B.L Rozovskiı, andI. Gyöngy. Degenerate parabolic stochastic partial differential equations (SPDEs) drivenby continuous noise were first investigated by N.V. Krylov and B.L. Rozovskiı in [KR82].These types of equations arise in the theory of non-linear filtering of continuous diffusionprocesses as the Zakai equation and as equations governing the inverse flow of continuous

87


diffusions. In [GGK14], the solvability of systems of linear SPDEs in Sobolev spaces wasproved by M. Gerencsér, I. Gyöngy, and N.V. Krylov, and a small gap in the proof ofthe main result of [KR82] was fixed. In Chapters 2, 3, and 4 of [Roz90], B.L. Rozovskiıoffers a unified presentation and extension of earlier results on the variational frameworkof linear stochastic evolution systems and SPDEs driven by continuous martingales (e.g.[Par75, KR77, KR79, KR82]). Our existence and uniqueness result on degenerate linearstochastic evolution equations driven by jump processes (Theorem 4.3.2) extends Theorem2 in Chapter-3-Section 2.2 of [Roz90] to include the important case of equations drivenby jump processes. It is also worth mentioning that the semigroup approach for non-degenerate SPDEs driven by Lévy processes is well-studied (see, e.g. [PZ07, PZ13]).

As a special case of (4.1.1), we will consider a system of SIDEs. Before introducingthe equation, let us describe our driving processes. Let η(dt, dz) be an integer-valuedrandom measure on (R+ × Z,B(R+) ⊗ Z) with predictable compensator πt(dz)dVt. Letη(dt, dz) = η(dt, dz) − πt(dz)dt be the martingale measure corresponding to η(dt, dz). Let(Z2,Z2) be a measurable space with RT -measurable family π2

t (dz) of sigma-finite randommeasures on Z. Let wϱ

t , t ≥ 0, ϱ ∈ N, be a sequence of continuous local uncorrelatedmartingales such that d⟨wϱ⟩t = dVt, for all ρ ∈ N. Let d1, d2 ∈ N. For convenience, weset (Z1,Z1) = (Z,Z) and π1

t = πt. We consider the d2-dimensional system of SIDEs on[0,T ] × Rd1 given by

dult =

((L1;l

t +L2;lt )ut + bi

t∂iult + cll

t ult(x) + f l

t

)dVt + (N lϱ

t ut + glϱt )dwϱ

t (4.1.2)

+

∫Z1

(Il

t,zult− + hl

t(z))η(dt, dz),

ul0 = φ

l, l ∈ 1, . . . , d2,

where for k ∈ 1, 2, l ∈ 1, . . . , d2, and ϕ ∈ C∞c (Rd1; Rd2),

Lk;lt ϕ(x) :=

12σ

k;iϱt (x)σk; jϱ

t (x)∂i jϕl(x) + σk;iϱ


l(x)∫Zk

((δll + ρ

k;llt (x, z)

) (ϕl(x + ζk(x, z)) − ϕl(x)

)− ζk;i

t (x, z)∂iϕl(x)

)πk

t (dz)

Nlϱt ϕ(x) := σ1;iϱ


t (x)ϕl(x), ϱ ∈ N,

Ilt,zϕ(x) := (δll + ρ

1;llt (x, z))ϕl(x + ζ1

t (x, z)) − ϕl(x),

and where δll is the Kronecker delta (i.e. δll = 1 if l = l and δll = 0 otherwise). Thesummation convention with respect to repeated indices is used here and below; summationover i is performed over the set 1, . . . , d1 and the summation over l, l is performed overthe set 1, . . . , d2. Without the noise term η(dt, dz) and integro-differential operators in L1

and L2, equation (4.1.2) has been well-studied (see, e.g. [KR82], [Roz90] (Chapter 3),

4.2. Degenerate linear stochastic evolution equations 89

and the recent paper [GGK14]). The choice to use the notation ζk(x, z) rather than thetraditional Hk(x, z) (as is used above) in this chapter due to the conflict of notation withour spaces.

Let (Hα(Rd1; Rd2))α∈R be the L2-Sobolev-scale. For each m ∈ N, using our theorem ondegenerate stochastic evolution equations discussed above, under suitable measurabilityand regularity conditions on the coefficients, initial condition, and free terms, we derivethe existence of a unique solution (ut)t≤T of (4.1.2) that is weakly càdlàg in Hm(Rd1; Rd2),strongly càdlàg in Hα(Rd1; Rd2) for all α < m, and belongs to L2(Ω × [0,T ],OT , dVtdP;Hm(Rd1; Rd2)).

Degenerate SIDEs of type (4.1.2) arise in the theory of non-linear filtering of semi-martingales as the Zakai equation and as the equations governing the inverse flow of jumpdiffusion processes. We constructed solutions of the above equation (with πt(dz) determin-istic and independent of time) using the method

This chapter is organized as follows. We derive our existence and uniqueness resultfor (4.1.1) in Section 4.2 and for (4.1.2) in Section 4.3.

4.2 Degenerate linear stochastic evolution equations

4.2.1 Basic notation and definitions

All vector spaces considered in this paper are assumed to have base field R. We alsoassume that all Hilbert spaces are separable. For a Hilbert space H, we denote by H∗ thedual of H and by B(H) the Borel sigma-algebra of H. Unless otherwise stated, the normand inner product of a Hilbert space H are denoted by | · |H and (·, ·)H, respectively. ForHilbert spaces H and U and a bounded linear map L : H → U, we denote by L∗ the Hilbertadjoint of L. Whenever we say that a map F from a sigma-finite measure space (S ,S, µ) toa Hilbert space H is S-measurable without specifying the sigma-algebra on H, we alwaysmean that F is S/B(H)-measurable. For any Hilbert space H and sigma-finite measurespace (S ,S, ν), we denote by L2(S ,S, µ; H) the linear space of all S-measurable functionsF : S → H such that

|F|L2(S ,S,ν;H) =

∫S|F(s)|2Hν(ds) < ∞,

where we identify functions F,G : S → H that are equal µ-almost-everywhere (ν-a.e.).The linear space L2(S ,S, ν; H) is a Hilbert space when endowed with the inner product

(F,G)L2(S ,S,ν;H) :=∫

S(F(s),G(s))Hν(ds).

We use the notation N = N(·, · · · , ·) below to denote a positive constant depending onlyon the quantities appearing in the parentheses. In a given context, the same letter is often


used to denote different constants depending on the same parameter. All the stochasticprocesses considered below are (at least) F-adapted unless explicitly stated otherwise.Furthermore, we will often drop the dependence on ω ∈ Ω for random quantities.

In this section, we consider a scale of Hilbert spaces (Hα)α∈R and a family of operators(Λα)α∈R satisfying the following properties:

• for all α, β ∈ R with β > α, Hβ is densely embedded in Hα;

• for all α, β, µ ∈ R with α < β < µ and all ε > 0, there is a constant N = N(α, β, µ, ε)such that

|v|β ≤ ε|v|µ + N |v|α, ∀v ∈ Hµ; (4.2.1)

• Λ0 = I; for all α, µ ∈ R, Λα : Hµ → Hµ−α is an isomorphism; for all α, β ∈ R,Λα+β = ΛαΛβ;

• for all α ∈ R, the inner product in Hα is given by (·, ·)α = (Λα·,Λα·)0;

• for all α > 0, the dual (Hα)∗ can be identified with H−α through the duality productgiven by

⟨u, v⟩α = ⟨u, v⟩Hα,H−α =(Λαu,Λ−αv

)0 , u ∈ Hα, v ∈ H−α;

• We assume that for every α ≥ 0, Λα is selfadjoint as an unbounded operator in H0

with domain Hα ⊆ H0: i.e. (Λαu, v)0 = (u,Λαv)0 for all u, v ∈ Hα.

Remark 4.2.1. It follows from the above properties that for all α ∈ R, the Hα norm is givenby |v|α = |Λαv|0, Λα is defined and linear on ∪β∈RHβ, Λ−α = (Λα)−1, and ΛαΛβ = ΛβΛα,for all β ∈ R. Moreover, for each α ≥ 0, if u ∈ Hα and v ∈ H0, then ⟨u, v⟩α = (u, v)0.

We will now describe our driving cylindrical martingale (Mt)t≥0 in (4.1.1) ) and theassociated stochastic integral. For a more thorough exposition, we refer to [MR99]. Let Ebe a locally convex quasi-complete topological vector space; all bounded closed subsetsof E are complete. Let E∗ be its topological dual. Denote by ⟨·, ·⟩E∗,E the canonical bilinearform (duality product) on E∗ × E. Assume that E∗ is weakly separable. Denote by L+(E)the space of symmetric non-negative definite forms Q from E∗ to E; that is, for all Q ∈L+(E), we have

⟨x,Qy⟩E∗,E = ⟨y,Qx⟩E∗,E, and ⟨x,Qx⟩E∗,E ≥ 0, ∀x, y ∈ E∗.

Recall that PT is the predictable sigma-algebra on Ω × [0,T ]. We say that a processQ : Ω×[0,T ]→ L+(E) isPT -measurable if ⟨y,Qtx⟩E∗,E isPT -measurable for all x, y ∈ E∗.


Assume that we are given a family of real-valued locally square integrable martingalesM = (Mt(y))y∈E∗)t≥0 indexed by E∗ and an increasing PT -measurable process Q : Ω ×[0,T ]→ L+(E) such that for all x, y ∈ E∗,

Mt(x)Mt(y) −∫ t

0⟨x,Qsy⟩E∗,EdVs, t ≥ 0,

is a local martingale.For each (ω, t) ∈ Ω × [0,T ], letHt = Hω,t be the Hilbert subspace of E defined as the

completion of Qω,tE∗ with respect to the inner product

(Qω,tx,Qω,ty

)Hω,t

:= ⟨x,Qω,ty⟩, x, y ∈ E∗.

It can be shown that for all (ω, t) ∈ Ω × [0,T ], E∗ is densely embedded into H∗t , the mapQt : E∗ → E can be extended to the Riesz isometry Qt : H∗t → Ht (still denoted Qt), andthe bilinear form ⟨x,Qty⟩E∗,E, x, y ∈ E∗, can be extended to ⟨x,Qty⟩H∗t ,Ht , x, y ∈ H∗t . Notethat for all x, y ∈ H∗t , we have (x, y)H∗t = ⟨x,Qty⟩H∗t ,Ht .

Let L2loc (Q) the space of all processes f such that ft ∈ H

∗t , dVtdP-a.e., ⟨ ft,Qty⟩H∗t ,Ht is

PT -measurable for all y ∈ E∗, and P-a.s.∫ T

0| ft|

2H∗t

dVt =

∫ T

0⟨ ft,Qt ft⟩H∗t ,HtdVt < ∞.

In [MR99], the stochastic integral of f ∈ L2loc(Q) against M, denoted It( f ) =

∫ t

0fsdMs,

t ≥ 0, was constructed and has the following properties: (It( f ))t≥0 is a locally squareintegrable martingale and P-a.s. for all t ∈ [0,T ]:

• for all y ∈ E∗, It(y) =∫ t

0ydMs = Mt(y) (recall that E∗ is embedded into allH∗s );

• for all g ∈ L2loc(Q).

⟨I( f ),I(g)⟩t =∫ t

0( fs, gs)H∗s dVs =

∫ t

0⟨ fs,Qsgs⟩H∗s ,HsdVs;

• for all bounded PT -measurable processes ϕ : Ω × [0,T ]→ R,∫ t

0ϕsdIs( f ) = It(ϕ f ) =

∫ t

0ϕs fsdMs.

For a Hilbert space H and (ω, t) ∈ Ω × [0,T ], denote by L2(H,H∗t ) the space of all


Hilbert-Schmidt operators Ψ : H → H∗t with norm and inner product given by

|Ψ|2L2(H,H∗t ) :=∞∑

n=1

|Ψhn|2H∗t, (Ψ, Ψ)L2(H,H∗t ) =

∞∑n=1

(Ψhn, Ψhn)H∗t , Ψ ∈ L2(H,H∗t ),

where (hn)n∈N is a complete orthogonal system in H. Denote by L2loc(H,Q) the space of

all processes Ψ such that Ψt ∈ L2(H,H∗t ), dVtdP-a.e., Ψth ∈ L2(Q), for each h ∈ H,, andP-a.s. ∫ T

0|Ψt|

2L2(H,H∗t )dVt < ∞.

For eachΨ ∈ L2loc (H,Q), we define the stochastic integral It(Ψ) =

∫ t

0ΨsdMs as the unique

H-valued càdlàg locally square integrable martingale such that P-a.s. for all t ∈ [0,T ] andh ∈ H,

(It(Ψ), h)H =

∫ t

0ΨshdMs.

For all Ψ, Ψ ∈ L2loc (H,Q) , we have that

|I· (Ψ) |2H −∫ ·

0|Ψs|

2L2(H,H∗s )dVs and (I·(Ψ),I·(Ψ))H −

∫ ·

0(Ψs, Ψs)L2(H,H∗s )dVs

are real-valued local martingales. Moreover, for all bounded PT -measurable H-valuedprocesses u : Ω × [0,T ]→ H, P-a.s. for all t ∈ [0,T ],∫ t

0usdIs(Ψ) =

∫ t

0usΨsHdMs,

where for a complete orthogonal system(en

s)

n∈N inH∗s ,

usΨsH :=∞∑

n=1

(Ψsus, en

s)H∗s

ens .

If H and Y are Hilbert spaces and L : H → Y is a bounded linear operator and Ψ ∈L2

loc(H,Q), then it follows that LIt(Ψ) = It(ΨL∗); indeed, for all y ∈ Y , we have

(LIt(Ψ), y)Y = (It(Ψ), L∗y)H =

∫ t

0ΨsL∗ydMs

and ΨL∗ ∈ L2loc(Y,Q).


4.2.2 Statement of main results

In this section, for µ ∈ (0, 1], we consider the linear stochastic evolution equation in thetriple (H−µ,H0,Hµ) given by

dut = (Ltut + ft) dVt + (Mtut− + gt) dMt, t ≤ T, (4.2.2)

u0 = φ,

where φ is an F0-measurable H0-valued random variable and Vt is a continuous non-decreasing process such that Vt ≤ C for all (ω, t) ∈ Ω × [0,T ], for some positive constantC. Let α ≥ µ be given. We assume that:

(1) the mapping L : Ω × [0,T ] × Hµ → H−µ is linear in Hµ, and for all v ∈ Hµ, Lv isRT/B(H−µ)-measurable; in addition, dVtdP-a.e., Ltv ∈ Hα−µ for all v ∈ Hα+µ;

(2) for dVtdP-almost-all (ω, t) ∈ Ω × [0,T ],Mω,t : Hµ → L2(H0,H∗ω,t) is linear, and forall v ∈ Hµ, ϕ ∈ H0, y′ ∈ E∗, ⟨(Mv)ϕ,Qty′⟩H∗t ,Ht is PT -measurable for all y′ ∈ E∗; inaddition, dVtdP-a.e,Mtv ∈ L2(Hα,H∗t ) for all v ∈ Hα+µ;

(3) the process f : Ω × [0,T ] → Hα−µ is RT/B(Hα−µ)-measurable and g ∈ L2loc (Hα,Q) ∩

L2loc(H

0,Q),

Let us introduce the following assumption for λ ∈ 0, α. Recall that (u, v)λ =(Λλu,Λλv

)0.

Assumption 4.2.1 (λ, µ). There are positive constants L and K and an RT -measurablefunction f : Ω × [0,T ]→ R such that the following conditions hold dVtdP-a.e.:

(1) for all v ∈ Hλ+µ,

2(Λµv,Λ−µLtv)λ + |Mtv|2L2(Hλ,H∗t ) ≤ L|v|2λ;

2(Λµv,Λ−µt Ltv + ft)λ + |Mtv + gt|2L2(Hλ,H∗t ) ≤ L|v|2λ + ft;

(2) for all v ∈ Hλ+µ,

|Ltv|λ−µ ≤ K|v|λ+µ, |Mtv|L2(Hλ,H∗t ) ≤ K|v|λ+µ;

(3)

| ft|2λ−µ + |gt|

2L2(Hλ,H∗t ) ≤ ft, E

∫ T

0ftdVt < ∞.

Let OT be the optional sigma-algebra on Ω × [0,T ]. For µ ∈ (0, 1]) and λ ∈ R+with λ ∈ 0, α, we denote byWλ,µ the space of all Hλ-valued strongly càdlàg processesv : Ω × [0,T ] → Hλ that belong to L2(Ω × [0,T ],OT , dVtdP; Hλ+µ). The following is ourdefinition of the solution of (4.2.2) and is standard in the variational theory or L2-theoryof stochastic evolution equations.


Definition 4.2.2. A process u ∈ W0,µ is said to be a solution of the stochastic evolutionequation (4.2.2) if P-a.s. for all t ∈ [0,T ]

utH−µ= u0 +

∫ t

0(Lsus + fs)dVs +

∫ t

0(Msus− + gs)dMs,

where H−µ= indicates that the equality holds in the H−µ. That is, P-a.s. for all t ∈ [0,T ] and

v ∈ Hµ,

(v, ut)0 = (v, u0) +∫ t

0⟨v,Lsus + fs⟩µdVs +

∫ t

0v(Msus− + gs)H0dMs.

Remark 4.2.3. In Definition 4.2.2, it is implicitly assumed that the integrals in (4.2.2) arewell-defined. Moreover, it is easy to check that if Assumption 4.2.1(0, µ) holds, then theintegrals in Definition 4.2.2 are well-defined.

In order to obtain estimates of the second moments of the supremum in t of the solutionof (4.2.2), in the Hα norm, we will need to impose the upcoming assumption. Beforeintroducing this assumption, we describe a few notational conventions. For two real-valued semimartingales Xt and Yt, we write P-a.s. dXt ≤ dYt if with probability 1, Xt−Xs ≤

Yt − Ys for any 0 ≤ s ≤ t ≤ T . For v ∈ Wλ,µ, we define

Mt (v) :=∫ t

0MsvsdMs, t ∈ [0,T ],

and denote by [M(v)]λ;t the quadratic variation process ofMt (v) in Hλ.

Assumption 4.2.2 (λ, µ). There is a positive constant L, a PT -measurable function g : Ω×[0,T ]→ R, and increasing adapted processes A, B : Ω×[0,T ]→ R with dAtdP ≤LdVtdP,dBtdP ≤gtdVtdP on PT such that the following conditions hold P-a.s.:

(1) for all v ∈ Wλ,µ,

(Λµvt,Λ−µLtvt)λdVt + d [M(v)]λ;t + 2vt−Mtvt−HλdMt ≤ |vt−|

2λdAt +Gt(v)dMt,

where G(v) ∈ L2loc (Q) satisfies |Gt(v)|H∗t dVt ≤ L|vt−|

2λdVt;

(2) for all v ∈ Wλ,µ,

2d[M (v) ,I (g)

]λ,t + 2vt−gtHλdMt ≤ |vt−|λdBt + Gt (v) dMt,

where G(v) ∈ L2loc (Q) satisfies |Gt(v)|H∗t dVt ≤ L|vt−|λgtdVt, and

E∫ T

0g2

t dVt < ∞.


Although Assumption 4.2.2(λ, µ) looks rather technical, it is satisfied for a large classof parabolic stochastic integro-differential equations (see Section 4.3) under what we con-sider to be reasonable assumptions.

Let T be the set of all stopping times τ ≤ T and T p be the set of all predictablestopping times τ ≤ T .

Theorem 4.2.4. Let µ ∈ (0, 1] and α ≥ µ. Let Assumption 4.2.1(λ, µ) hold for λ ∈ 0, αand assume that E

[|φ|2α

]< ∞.

(1) Then there exists a unique solution u = (ut)t≤T of (4.2.2) such that for any α′ < α, uis an Hα′-valued strongly càdlàg process and there is a constant N = N(L,K,C) suchthat

E[supt≤T|ut|

2α−µ

]+ sup

τ∈T

E[|uτ|2α

]+ E

∫ T

0|us|

2α dVs ≤ N

(E

[|φ|2α

]+ E

∫ T

0ftdVt

).

Moreover, for all p ∈ (0, 2) and α′ < α, there is a constant N = N(L,K,C, p, α′) suchthat

E[supt≤T|ut|

pα′

]≤ NE

(|φ|2α + ∫ T

0ftdVt

) p2 .

(2) If, in addition, Assumption 4.2.2(λ, µ) holds for λ ∈ 0, α, then u is an Hα-valuedweakly càdlàg process and there is a constant N = N(L,K,C) such that

E[supt≤T|ut|

2α

]≤ NE

[|φ|2α +

∫ T

0( ft + g2

t )dVt

].

Remark 4.2.5. If V is an arbitrary continuous increasing adapted process, then Theo-rem 4.2.4 can be applied locally by considering VC

t = Vt∧τC , t ∈ [0,T ], with τC = inf(t ∈ [0,T ] : Vt ≥ C) ∧ T.


We will construct a sequence of approximations in Wα,µT of the solution of (4.2.2) by

solving in the triple (H−µ,H0,Hµ) the equation

dut =(Ln

t ut + ft)

dVt + (Mtut− + gt) dMt, t ≤ T, (4.2.3)

u0 = φ,

where Lnt = Lt −

1n (Λµ)2. In order to apply the foundational theorems on stochastic evolu-

tion equations with jumps established in [GK81] and [Gyö82], it is convenient for us first


to consider the following equation in the triple (H−µ,H0,Hµ) :

dvt =

(ΛαLtΛ

−αvt −1n

(Λµ)2vt + Λα ft

)dVt +

(MtΛ

−αvt−(Λα)∗ + gt(Λα)∗)

dMt, t ≤ T,(4.2.4)

v0 = Λαφ.

The solutions of (4.2.3) and (4.2.4) are to be understood following Definition 4.2.2.

Lemma 4.2.6. Let µ ∈ (0, 1] and α ≥ µ. Let Assumption 4.2.1(α, µ) hold and assume thatE

[|φ|2α

]< ∞.

(1) For each n ∈ N, there is a unique solution vn = (vnt )t≤T of (4.2.3), and there is a

constant N = N(L,K,C) independent of n such that

supτ∈T

E[|vnτ|

20

]+ E

∫ T

0|vn

t |20dVt +

1n

E∫ T

0|vn

t |2µdVt ≤ NE

[|φ|2α +

∫ T

0ftdVt

]. (4.2.5)

Moreover, for all p ∈ (0, 2), there is a constant N = N(L,K,T, p)

E[supt≤T|vn

t |p0

]≤ NE

(|φ|2α + ∫ T

0ftdVt

) p2 . (4.2.6)

(2) If, in addition, Assumption 4.2.2(α, µ) holds, then there is a constant N = N(L,K,C)such that

E[supt≤T|vn

t |20

]≤ NE

[|φ|2α +

∫ T

0( ft + g2

t )dVt

]. (4.2.7)

Proof. (1) For each (ω, t) ∈ Ω × [0,T ] and n ∈ N, let

Lαt v = ΛαLtΛ−αv, L

α,nt v = Lαt v −

1n

(Λµ)2 v, Mαt v =MtΛ

−αv(Λα)∗.

Using basic properties of the spaces (Hα)α∈R and the operators (Λα)α∈R, dVtdP-a.e. for allv ∈ Hµ, we have

2⟨v,Lαt v⟩µ = 2(Λµv,Λ−µΛαLtΛ−αv)0 = 2(ΛµΛ−αv,Λ−µLtΛ

−αv)α,

2⟨v, (Λµ)2v⟩µ = 2(Λµv,Λµv)0 = |v|2µ,

and

∣∣∣Mαt v

∣∣∣2L2(H0,H∗t )

=

∞∑k=1

|Λα(MtΛ−αv)∗en

t |2H0 =

∞∑k=1

|(MtΛ−αv)∗en

t |2Hα

=

∞∑k=1

|MtΛ−αvhk|2

H∗t=

∣∣∣Mαt v

∣∣∣2L2(Hα,H∗t )

,


where (ekt )k∈N,and (hk)k∈N are orthonormal basis ofHt and Hα, respectively. It follows from

Assumption 4.2.1(α, µ) that dVtdP-a.e. for all v ∈ Hµ, we have

|Lα,nt v|−µ ≤

(K +

1n

)|v|µ,

∣∣∣Mαt v

∣∣∣L2(H0,H∗t )

≤ K|v|µ,

and

2⟨v,Lα,nt v + Λα ft⟩µ +∣∣∣Mα

t v + gt(Λα)∗∣∣∣2L2(H0,H∗t )

≤ −2n|v|2µ + L|v|20 + ft. (4.2.8)

In [Gyö82], the variational theory for monotone stochastic evolution equations drivenby locally square integrable Hilbert-space-valued martingales was derived; it is worthmentioning that the càdlàg version of the variational solution in the pivot space and theuniqueness of the solution was obtained using Theorem 2 in [GK81]. The theorems andproofs given in [Gyö82] continue to hold for equations driven by the cylindrical martin-gales we consider in this paper. Therefore, by Theorems 2.9 and 2.10 in [Gyö82], for everyn ∈ N, there exists a unique solution vn = (vn

t )t≤T of the stochastic evolution equation givenby

vnt = Λ

αφ +

∫ t

0(Lα,ns vn

s + Λα fs)dVs +

∫ t

0(Mα

s vns− + Λ

αgs)dMs, t ≤ T.

Furthermore, by virtue of Theorem 4.1 in [Gyö82], there is a constant N(n) = N(n, L,K,C) such that

E[supt≤T|vn

t |20

]+ E

∫ T

0|vn

t |2µdVt ≤ N(n)E

[|φ|2α +

∫ T

0ftdVt

]. (4.2.9)

We will now use our assumptions to obtain bounds of the solutions vnt , n ∈ N, in the

H0-norm independent of n. Applying Theorem 2 in [GK81], P-a.s. for all t ∈ [0,T ], wehave

|vnt |

20 = |φ|

2α +

∫ t

02⟨vn

s ,Lα,nt vn

s + Λα fs⟩µdVs +

[M

]0,t+ mt, (4.2.10)

where (Mt)t≤T and (mt)t≤T are local martingales given by

Mt :=∫ t

0(Mα

s vns− + Λ

αgs)dMs, mt := 2∫ t

0vn

s−(Mαs vn

s− + Λαgs)H0dMs.

Thus, taking the expectation of (4.2.10) and making use of Assumption 4.2.1(α, µ), (4.2.8),and (4.2.9), we find that for all τ ∈ T ,

E[|vnτ|

20

]≤ E

[|φ|2α

]+ E

∫ τ

0

(2⟨vn

t ,Lα,nt vn

t + Λα ft⟩µ + |M

αt v + Λαgt|

2L2(H0,Ht)

)dVt

≤ E[|φ|2α +

∫ τ

0

(L|vn

t |20 −

2n|vn

t |2µ + ft

)dVt

].


This implies that for any τ ∈ T p,

E[|vnτ−|

20

]+

2n

E∫ τ

0|vn

t |2µdVt ≤ E

[|φ|2α +

∫ τ

0

(L|vn

s |20 + fs

)dVs

].

By virtue of Lemmas 2 and 3 in [GM83], we deduce that there is a constant N = N(L,K,C)such that for any τ ∈ T ,

E[|vnτ|

20

]+

1n

E∫ τ

0|vn

t |2µdVt ≤ NE

[|φ|2α +

∫ τ

0fsdVs

],

which implies that (4.2.5) holds since Vt is uniformly bounded by the constant C. Finally,owing to Corollary II in [Len77], we have (4.2.6).

(2) Using Assumption 4.2.2(α, µ) and estimating (4.2.10), we get that P-a.s.

d|vnt |

20 ≤ |φ|

2α + |v

nt−|

2λdAt + |vn

t−|λdBt + (Gt(vn) + Gt (vn))dMt.

Moreover, for any τ ∈ T , we obtain

E[supt≤τ|vn

t |20

]≤ NE

[|φ|2α +

∫ τ

0|vn

t |20dVt +

∫ τ

0( ft + g2

t )dVt + supt≤τ|ln

t |

],

where lnt :=

∫ t

0(Gs(vn)+Gs(vn))dMs.Moreover, by the Burkholder-Davis-Gundy inequality

and Young’s inequality, we have

E supt≤τ|ln

t | ≤ NE

(∫ τ

0(|vn

t−|4α + |v

nt−|

2αg2

t )dVt

) 12 ≤ NE

supt≤τ|vn

t |0

(∫ τ

0(|vn

t−|2α + g2

t )dVt

) 12

≤1

4NE

[supt≤τ|vn

t |20

]+ NE

∫ τ

0(|vn

t |2α + g2

t )dVt,

from which estimate (4.2.7) follows.

Proof of Theorem 4.2.4 . (1) Let vn = (vnt )t≤T be the unique solution of (4.2.4) constructed

in Lemma 4.2.6. Since vn ∈ W0,µ, it follows that un := Λ−αvn ∈ Wα,µ ⊆ W0,µ is a solutionof (4.2.3) in the triple (H−µ,H0,Hµ).

We will first show that (un)n∈N is Cauchy in H0. Letting un,m = un−um, for all n,m ∈ N,we have

un,mt =

∫ t

0(Ln

t unt − L

mt um

t )dVt +

∫ t

0Mtun,mdMt, t ≤ T.

Applying Theorem 2 in [GK81], we obtain that P-a.s. for all t ∈ [0,T ],

|unt − um

t |20 =

∫ t

02⟨un

s − ums ,L

nsu

ns − L

ms um

s ⟩µ,0dVs + [Mn,m]t + ηn,mt (4.2.11)


where (Mn,mt )t≤T and (ηn,m

t )t≤T are local martingales given by

Mn,mt :=

∫ t

0Ms(un

s− − ums−)dMs, ηm,n

t := 2∫ t

0(un

s− − ums−)Ms(un

s− − ums−)H0dMs.

Assumption 4.2.1(0, µ) (1) implies that for any τ ∈ T ,

E[|un,mτ |

20

]≤ E

∫ τ

0

(2⟨un,m

s ,Lnsu

n,ms ⟩µ + |Mtun,m

s |2L2(H0,H∗t )

)dVs

≤ E∫ τ

0

(L|un,m

s |20 +

1n|un

s |2µ +

1m|um

s |2µ

)dVs,

and hence for any τ ∈ T p, we have

E[|un.mτ− |

20

]≤ E

∫ τ

0

(L|un.m

s |20 +

1n|un

s |2µ +

1m|um

s |2µ

)dVs.

By virtue of Lemmas 2 and 4 in [GM83] and (4.2.5) (noting that |unt |0 = |Λ

−αvnt |0 ≤ N|vn

t |0),there is a constant N such that for any τ ∈ T ,

E[|un,mτ |

20

]≤ NE

∫ τ

0

(1n|un

s |2µ +

1m|um

s |2µ

)dVs ≤ N

(1n+

1m

)E

[|φ|2α +

∫ T

0ftdVt

].

Using Corollary II in [Len77], we have that for all p ∈ (0, 2), there is a constant N suchthat

E[supt≤T|un,m

t |p0

]≤ N

(1n+

1m

) p2[E

[|φ|2α +

∫ T

0ftdVt

]] p2

.

Therefore,

limn,m→∞

[supτ∈T

E[|un,mτ |

20

]+ E

[supt≤T|un,m

t |p0

]]= 0, (4.2.12)

and hence there exists a strongly càdlàg H0-valued process u = (ut)t≤T such that

dP − limn→∞

supt≤T|ut − un

t |0 = 0. (4.2.13)

Since for all n, un is solution of (4.2.3), we have that P-a.s. for all t ∈ [0,T ] andϕ ∈ Hµ,

(ϕ, unt )0 = (ϕ, φ)0 +

∫ t

0⟨ϕ,Ln

suns + fs⟩µ,0dVs +

∫ t

0ϕ(Msun

s− + gs)H0dMs. (4.2.14)

Owing to (4.2.13), we know that the left-hand-side of (4.2.14) converges P-a.s. for allt ∈ [0,T ] to (ϕ, ut)0 as n tends to infinity. Our aim, of course, is to pass to the limit as ntends to infinity on the right-hand-side.


This can be done quite simply when α > µ. Indeed, owing to the interpolation inequal-ity (4.2.1), for all ε > 0, α′ < α, and p ∈ (0, 2), there is a constant N = N(α, α′, ε, p) suchthat

supt≤T|un,m

t |pα′ ≤ ε sup

t≤T|un,m

t |pα + N sup

t≤T|un,m

t |p0 . (4.2.15)

Since |unt |α = |Λ

−αvnt |α ≤ N|vn

t |0, by (4.2.7) and (4.2.15), we have that for all α′ < α andp ∈ (0, 2),

E[supt≤T|un,m

t |pα′

]≤ εNE

(|φ|2α + ∫ T

0ftdVt

) p2 + NE

[supt≤T|un,m

t |p0

]. (4.2.16)

Using (4.2.12) and passing to the limit as n and m tend to infinity on both sides of (4.2.16),and then taking ε ↓ 0, we get that for all α′ < α and p ∈ (0, 2),

limn,m→∞

E[supt≤T|un,m

t |pα′

]= 0.

Combining the above results, we conclude that for any α′ < α, u is an Hα′-valued stronglycàdlàg process and

dP − limn→∞

supt≤T|ut − un

t |α′ = 0. (4.2.17)

In particular, if α > µ, then taking α′ > µ in (4.2.17) and appealing to Assumption4.2.1(0, µ) (2), (4.2.5), and the identity,

⟨ϕ,Λ2µuns⟩µ = (Λµϕ,Λµun

s)0,

we can take the limit as n tends to infinity on the right-hand-side of (4.2.14) by the domi-nated convergence theorem to conclude that u is a solution of (4.2.2).

The case α = µ must be handled with weak convergence. Let

S (OT ) = (Ω × [0,T ],OT , dVtdP) and S (PT ) = (Ω × [0,T ],PT , dVtdP),

where Vt =: Vt+t. It follows from (4.2.5) that there exists a subsequence (unk)k∈N of (un)n∈N

that converges weakly in L2(S (OT ); Hµ) to some u ∈ L2(S (OT ); Hµ) which satisfies

E∫

]0,T ]|ut|

2µdVt ≤ NE

[|φ|2µ +

∫ T

0ftdVt

].

For any ϕ ∈ H0 and bounded predictable process ξt, we have

limk→∞

E∫ T

0ξt⟨ϕ, u

nkt ⟩µdVt = lim

k→∞E

∫ T

0ξt(u

nkt , ϕ)0dVt = E

∫ T

0ξt(ut, ϕ)0dVt,


and hence u = u in L2(S (OT ); Hµ) and unk converges to u weakly in L2(S (OT ); Hµ) as ktends to infinity. Define unk

− = (unkt−)t≤T and u− = (ut−)t≤T , where the limits are taken in

H0. By repeating the argument above, we conclude that unk− converges to u− weakly in

L2(S (OT ); Hµ) as k tends to infinity.Fix ϕ ∈ Hµ and a PT -measurable process (ξt)t≤T bounded by the constant K. Define

the linear functionals ΦL : L2(S (OT ); Hµ)→ R and ΦM : L2(S (PT ); Hµ)→ R by

ΦL(v) = E∫

]0,T ]ξt

∫]0,t]⟨ϕ,Lsvs⟩µ,0dVsdVt, ∀v ∈ L2(S (OT ); Hµ)

andΦM(v) = E

∫]0,T ]

ξt

∫]0,t]ϕMsvsH0dMsdVt, ∀v ∈ L2(S (PT ); Hµ),

respectively. Owing to Assumption 4.2.1(0, µ) (2), the Burkholder-Davis-Gundy inequal-ity, and the fact that (Vt)t≤T is uniformly bounded by the constant C, there is a constantN = N(K,C) such that

|ΦL(v)| ≤ N |ϕ|µ

(E

∫ T

0|vt|

2µdVt

) 12

, ∀v ∈ L2(S (OT ); Hµ),

and

|ΦM(v)| ≤ N|ϕ|µ

(E

∫]0,T ]|vs|

2µdVs

) 12

, ∀v ∈ L2(S (PT ); Hµ).

This implies that ΦL is a continuous linear functional on L2(S (OT ); Hµ) and ΦM is a con-tinuous linear functional on L2(S (PT ); Hµ), and hence that

limk→∞ΦL(unk) = ΦL(u), lim

k→∞ΦL(unk

− ) = ΦM(u−). (4.2.18)

For each k, we have that

E∫ T

0ξt(ϕ, u

nkt )0dVt = E

∫ T

0ξt(ϕ, φ)0dVt + E

∫ T

0ξt

∫ t

0⟨ϕ,Lnk

s unks + fs⟩µ,0dVsdVt(4.2.19)

+ E∫ T

0ξt

∫ t

0ϕ(Msun

s− + gs)H0dMsdVt.

Passing to the limit as k tends to infinity on both sides of (4.2.19) using (4.2.18) and

⟨ϕ,Λ2µunks ⟩µ = (Λµϕ,Λµunk

s )0,


we obtain that dVtdP-a.e.

(ϕ, ut)0 = (ϕ, φ)0 +

∫]0,t]⟨ϕ, (Lsus + fs)⟩µdVs +

∫]0,t]ϕ(Msus− + gs)H0dMs, t ≤ T.

Therefore, for all α ≥ µ, u is a solution of (4.2.2).We will now show that

supτ∈T

E[|uτ|2α

]≤ NE

[|φ|2α +

∫ T

0ftdVt

]. (4.2.20)

Let (hk)k∈N be a complete orthonormal basis in Hα such that the linear subspace spannedby (hk)k∈N is dense in H2α. Owing to (4.2.5), for all m ≥ 1 and τ ∈ T ,

E m∑

k=1

|(unτ,Λ

2αhk)0|2

= E m∑

k=1

|(unτ, h

k)α|2 ≤ E

[|unτ|

2α

]≤ NE

[|φ|2α +

∫ T

0ftdVt

].

Applying Fatou’s lemma first in n and then in m, we have that for all τ ∈ T ,

E ∞∑

k=1

|(uτ,Λ2αhk)0|2

≤ NE[|φ|2α +

∫ T

0ftdVt

].

Hence, for all t ∈ [0,T ], P-a.s. vt =∑

k(ut,Λ2αhk)0hk ∈ Hα. Since the linear subspace

spanned by (Λ2αhk)k∈N is dense in H0 and for all t ∈ [0,T ], P-a.s., (ut − vt,Λ2αhk)0 = 0, for

all k ∈ N, it follows that P-a.s. for all τ ∈ T , uτ = v and

E[|uτ|2α

]= E

∞∑k=1

∣∣∣(uτ,Λ2αhk)0

∣∣∣2 ≤ NE[|φ|2α +

∫ T

0ftdVt

],

which proves (4.2.20).Estimating (4.2.2) directly in the Hα−µ-norm, we easily derive that

E[supt≤T|ut|

2α−µ

]≤ NE

[|φ|2α +

∫ T

0fsdVt

].

If (vt)t≤T be another solution of (4.2.2), then by Theorem 2 in [GK81] and Assumption4.2.1(0, µ)(1), P-a.s. for all t ∈ [0,T ], we have

|ut − vt|20 ≤ L

∫ t

0|us − vs|

20dVs + mt,

where (mt)t≤T is a local martingale with m0 = 0, and hence applying Lemmas 2 and 4 in


[GM83], we get

P(supt≤T|ut − vt|0 > 0

)= 0,

which implies that (ut)t≤T is the unique solution of (4.2.2). This completes the proof ofpart (1).

(2) Estimating (4.2.11) using Assumption 4.2.2(0, µ), we get that P-a.s.

d|un,mt |

20 ≤ |φ|

20 +

(1n+

1m

)|un,m|20dVt + |un,m

t− |2λdAt + |un,m

t− |λdBt + (Gt(un,m) + Gt (un,m))dMt.

Then estimating the stochastic integrand as in the proof of part (2) of Lemma 4.2.6, forany τ ∈ T , we get

E[supt≤τ|un.m

t |20

]≤ NE

∫ τ

0

(|un.m

s |20 +

1n|un

s |2µ +

1m|um

s |2µ))

dVs,

and hence by Gronwall’s lemma and Lemma 4.2.6(2),

E[supt≤τ|un.m

t |20

]≤ N

(1n+

1m

)E

[|φ|2α +

∫]0,T ]

(ft + g2

t

)dVt

]Thus,

limn,m→∞

E[supt≤τ|un.m

t |20

]= 0.

Let (hk)k∈N be a complete orthonormal basis Hα such that the linear subspace spanned by(hk)k∈N is dense in H2α. Owing to (4.2.7), for all m ≥ 1 and τ ∈ T ,

Esup

t≤T

m∑k=1

(unt , h

k)α

= Esup

t≤T

m∑k=1

|(unt ,Λ

2αhk)0|2

≤ E[supt≤T|un

t |2α

]≤ NE

[|φ|2α +

∫]0,T ]

( ft + g2t )dVt

].

Applying Fatou’s lemma first in n and then in m, we have that

Esup

t≤T

∞∑k=1

|(ut,Λ2αhk)0|

2

≤ NE[|φ|2α +

∫]0,T ]

( ft + g2t )dVt

].

Thus, v =∑

k(u,Λ2αhk)0hk is an Hα-valued weakly càdlàg process. Since the linear sub-space spanned on (Λ2αhk)k∈N is dense in H0 and

(ut − vt,Λ

2αhk)

0= 0, for all k ∈ N, it

follows that P-a.s. for all t ∈ [0,T ], ut = vt and

E[supt≤T|ut|

2α

]= E

supt≤T

∞∑k=1

|(ut,Λ2αhk)0|

2

≤ NE[|φ|2α +

∫]0,T ]

( ft + g2t )dVt

].

4.3. The L2-Sobolev theory for degenerate SIDEs 104

4.3 The L2-Sobolev theory for degenerate SIDEs

4.3.1 Statement of main results

In this section, we consider the d2-dimensional system of SIDEs on [0,T ] × Rd1 given by

dult =

((L1;l

t +L2;lt )ut + bi

t∂iult + cll

t ult + f l

t

)dVt +

(N

lϱt ut + glϱ

t

)dwϱ

t (4.3.1)

+

∫Z1

(Il

t,zult− + hl

t(z))η(dt, dz),

ul0 = φ

l, l ∈ 1, . . . , d2,

where for k ∈ 1, 2, l ∈ 1, . . . , d2, and ϕ ∈ C∞c (Rd1; Rd2),

Lk;lt ϕ(x) :=

12σ

k;iϱt (x)σk; jϱ

t (x)∂i jϕl(x) + σk;iϱ


l(x)

+

∫Zk

((δll + ρ

k;llt (x, z)

) (ϕl(x + ζk

t (x, z)) − ϕl(x))− ζk;i

t (x, z)∂iϕl(x)

)πk

t (dz)

Nlϱt ϕ(x) := σ1;iϱ


t (x)ϕl(x), ϱ ∈ N,

Ilt,zϕ(x) :=

(δll + ρ

1;llt (x, z)

)ϕl(x + ζ1

t (x, z)) − ϕl(x).

We assume that

σkt (x) = (σk;iϱ

ω,t (x))1≤i≤d1, ϱ∈N, bt(x) = (biω,t(x))1≤i≤d1 , ct(x) = (cll

ω,t(x))1≤l,l≤d2 ,


ω,t (x))1≤l,l≤d2, ϱ∈N, ft(x) = ( f iω,t(x))1≤i≤d2 , gt(x) = (giϱ

ω,t(x))1≤i≤d2, ϱ∈N

are random fields on Ω × [0,T ] × Rd1 that are RT ⊗ B(Rd1)-measurable. Moreover, weassume that

ζ1t (x, z) = (ζ1;i

ω,t(x, z))1≤i≤d1 , ρ1t (x, z) = (ρ1;ll

ω,t (x, z))1≤l,l≤d2 , hiω,t(x, z))1≤i≤d2 ,

are random fields on Ω × [0,T ] × Rd1 × Z1 that are PT ⊗ B(Rd1) ⊗Z1-measurable and

ζ2t (x, z) = (ζ2;i

ω,t(x, z))1≤i≤d1 , ρ2t (x, z) = (ρ2;ll

ω,t (x, z))1≤l,l≤d2 ,

are random fields on Ω× [0,T ]×Rd1 ×Z2 that are RT ⊗B(Rd1)⊗Z2-measurable. We alsoassume that there is a constant C such that Vt ≤ C for all (ω, t) ∈ Ω × [0,T ].

Let us introduce the following assumption for an integer m ≥ 1 and a real numberα1 ∈ (1, 2). We remind the reader that the norms and seminorms | · |0 = [·]0 and | · |β,


β ∈ (0, 1], were defined in the beginning of Section 2.2.Let us introduce the following assumption for m ∈ N and a real number β ∈ [0, 2].

Assumption 4.3.1 (m, d2). Let N0 be a positive constant.

(1) For all (ω, t, x) ∈ Ω × [0,T ] × Rd1 , the derivatives in x of the random fields bt, ct, σ2t ,

and υ2t up to order m and σk

t and υkt , k ∈ 1, 2, up to order m + 1 exist, and for all

x ∈ Rd1 ,

max|γ|≤m

(|∂γbt(x)| + |∂γct(x)| + |∂γ∇σ1

t (x)| + |∂γσ2t (x)| + |∂γ∇υ1

t (x)| + |∂γυ2t (x)|

)≤ N0.

(2) For each k ∈ 1, 2 and all (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × Zk, the derivatives in x ofthe random field ζk

t (z) up to order m exist, and for all x ∈ Rd1 ,

max|γ|≤m|∂γζ1

t (x, z) | +max|γ|=m

[∂γζ1

t (·, z)]β2≤ K1

t (z) , ∀z ∈ Z1,

max|γ|≤m|∂γζ2

t (x, z) | ≤ K2t (z), ∀z ∈ Z2,

where K1t (resp. K2

t ) are PT ⊗Z1 (resp. PT ⊗Z

1) -measurable processes satisfying

supz∈Zk

Kkt (z) +

∫Z1

K1t (z)βπ1

t (dz) +∫

Z2K2

t (z)2π2(dz) ≤ N0.

(3) There is a constant η < 1 such that for each k ∈ 1, 2 on the set all (ω, t, x, z) ∈ Ω ×[0,T ] × Rd1 × Zk for which |∇ζk

t (x, z)| > η,∣∣∣∣(Id1 + ∇ζkt (x, z)

)−1∣∣∣∣ ≤ N0.

(4) For each k ∈ 1, 2 and all (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × Zk, the derivatives in x ofthe random field ρk

t (z) up to order m exist, and for all x ∈ Rd1 ,

max|γ|≤m|∂γρ1

t (x, z)| +max|γ|=m

[ρ1

t (·, z)]β2≤ l1

t (z),

max|γ|≤m|∂γρ2

t (x, z)| ≤ l2t (z) ,

where l1 (resp. l2) is PT ⊗Z1 (resp. PT ⊗Z

2) -measurable function satisfying∫Z1

l1t (z)2π1

t (dz) +∫

Z2l2t (z)2π2(dz) ≤ N0.

Let L2 = L2(Rd1 ,B(Rd1), ν; R), where ν (differential is denoted by dx) is the Lebesguemeasure. Let S = S(Rd1) be the Schwartz space of rapidly decreasing functions on Rd1 .


The Fourier transform of an element v ∈ S is defined by

v(ξ) = F v(ξ) =∫

Rd1

v(x)e−i2πξ·xdx, ξ ∈ Rd1 .

We denote by F −1 its inverse. Denote the space of tempered distributions by S′, the dualof S.

Let ∆ :=∑d1

i=1 ∂2i be the Laplace operator on Rd1 . For α ∈ R, we define the Sobolev

scale

Hα(Rd1; Rd)

=

v = (vi)1≤i≤d : vi ∈ S′ and

(1 + 4π2 |ξ|2

)α/2vi ∈ L2(Rd1), ∀i ∈ 1, . . . , d

=

v = (vi)1≤i≤d : vi ∈ S′ and (I − ∆)

α2 vi ∈ L2(Rd1), ∀i ∈ 1, . . . , d

with the norm and inner product given by

∥v∥α,d =

d∑i=1

∣∣∣∣(1 + 4π2 |ξ|2)α/2

vi∣∣∣∣2L2

1/2

=

d∑i=1

∣∣∣(I − ∆)α/2 vi∣∣∣2L2

1/2

and

(v, u)α,d =d∑

i=1

((I − ∆)α/2 vi, (I − ∆)α/2 ui

)L2, ∀u, v ∈ Hα(Rd1; Rd),

where(I − ∆)α/2 vi = F −1

((1 + 4π2 |ξ|2

)α/2vi).

It is well-known that C∞c (Rd1; Rd) is dense in Hα(Rd1; Rd) for all α ∈ R. For v ∈ H1(Rd1; Rd)and u ∈ H−1(Rd1; Rd), we let

⟨v, u⟩1,d = (Λ1v,Λ−1u)0,d,

and identify the dual of H1(Rd1; Rd) with H−1(Rd1; Rd) through this bilinear form. More-over, all of the properties imposed in Section 4.2 for the abstract family of spaces (Hα)α∈Rand operators (Λα)α∈R hold for the Sobolev scale. We refer the reader to [Tri10] for moredetails about the Sobolev scale (see the references therein as well).

For each α ∈ R, let Hα(Rd1; Rd;F0) be the space of all F0-measurable Hα(Rd1; Rd)-valued random variables φ satisfying E

[∥φ∥2α

]< ∞.

Let Hα(Rd1; Rd2) be the space of all Hα(Rd1; Rd2)-valued RT -measurable processes f :


Ω × [0,T ]→ Hα(Rd1; Rd2) such that

E∫ T

0∥ ft∥

2αdVt < ∞.

Let Hα(Rd1; ℓ2(Rd)) be the space of all sequences of Hα(Rd1; Rd)-valuedPT -measurableprocesses g = (gϱ)ϱ∈N, gϱ : Ω × [0,T ]→ Hα(Rd1; Rd), satisfying

E∫ T

0∥gt∥

2αdVt = E

∫ T

0

∑ρ∈N

∥gϱt ∥2αdVt < ∞.

Let Hα(Rd1; Rd; π1) be the space of all Hα(Rd1; Rd)-valued PT ⊗Z1-measurable processes

h : Ω × [0,T ] ×Z1 → Hα(Rd1; Rd) such that

E∫ T

0

∫Z1∥ht (z) ∥2απ

1t (dz)dVt < ∞..

For each α ∈ R, we set Hα = Hα(Rd1; Rd2), Hα(F0), Hα = Hα(Rd1; Rd2), Hα(ℓ2) =Hα(Rd1; ℓ2(Rd2)), Hα(π1) = Hα(Rd1; Rd2; π1), and ∥ · ∥α = ∥ · ∥α,d2 , (·, ·)α = (·, ·)α,d2 , ⟨·, ·⟩1 =

⟨·, ·⟩1,d2 . We also set C∞c = C∞c (Rd1; Rd).

Definition 4.3.1. Let φ ∈ H0(F0), f ∈ H−1, g ∈ H0(ℓ2), and h ∈ H0(π1). An H0-valuedstrongly càdlàg process u = (ut)t≤T is said to be a solution of the SIDE (4.3.1) if u ∈L2(Ω × [0,T ],OT , dVtdP; H1) and P-a.s. for all t ∈ [0,T ],

utH−1

= φ +

∫ t

0

((L1;l

s +L2;ls )us + bi

s∂iuls + cll

s uls + f l

s

)dVs +

∫ t

0

(N lϱ

s us + glϱs

)dwϱ

s

+

∫ t

0

∫Z1

(Il

s,zuls− + hl

s(z))η(ds, dz),

where H−1

= indicates that the equality holds in the H−1. That is, P-a.s. for all t ∈ [0,T ] andv ∈ H1,

(v, ut)0 = (v, u0) +∫ t

0⟨v, (L1

s +L2s)us + bi

s∂ius + csus + fs⟩1dVs

+

∫ t

0

(v,

(N lϱ

s us + glϱs

))0

dwϱs +

∫ t

0

∫Z1

(v,

(Il

s,zuls− + hl

s(z)))η(ds, dz).

The coming theorem is our existence result for equation (4.3.1). In the next section,we prove this theorem by appealing to Theorem 4.2.4.

Theorem 4.3.2. Let Assumption 4.3.1(m, d2) hold for m ∈ N and a real number β ∈ [0, 2].Then for every φ ∈ Hm(F0), f ∈ Hm, g ∈ Hm+1(ℓ2), h ∈ Hm+ β2 (π1), and there exists aunique solution u = (ut)t≤T of (4.3.1) that is weakly càdlàg as an Hm-valued process and


strongly càdlàg as an Hα′-valued process for any α′ < m. Moreover, there is a constantN = N(d1, d2,N0,m, η, β) such that

E[supt≤T∥ut∥

2m

]≤ NE

[∥φ∥2m +

∫ T

0

(∥ ft∥

2m + ∥gt∥

2m+1 +

∫Z1∥ht(z)∥2

m+ β2π1

t (dz))

dVt

].

The following corollary can be proved in the same way as Corollary 1 (with p = 2) inChapter 4, Section 2.2 of [Roz90] by making use of the Sobolev embedding theorem.

Corollary 4.3.3. Let Assumption 4.3.1(m, d2) hold for an integer m > d12 and a real number

β ∈ [0, 2]. Then the solution (ut)t≤T of (4.3.1) has a version with the following properties:

(1) for every x ∈ Rd1 , ut(x) is a OT -measurable Rd2-valued process;(2) for β = (2m − d1)/2 and for all ω ∈ Ω, u ∈ D([0,T ];Cβloc(R

d1; Rd2));(3) it possess all the properties mentioned in Theorem 4.3.2;(4) for each bounded subset Q ⊆ Rd,

E[supt≤T|ut|

2β;Q;Rd2

]+ E

∫]0,T ]∥ut∥

2β;Q;Rd2

dVt

≤ NE[∥φ∥2m +

∫ T

0

(∥ ft∥

2m + ∥gt∥

2m+1 +

∫Z1∥ht(z)∥2

m+ β2π1

t (dz))

dVt

],

where N = N(d1, d2,N0,m, η, β) is a constant;(5) if (u1

t )t≤T and (u2t )t≤T are solutions of (4.3.1) possessing properties (1) and (2), then

P sup

t≤T,x∈Rd1

|u1t (x) − u2

t (x)| > 0

= 0.

Remark 4.3.4. If m > α1∨α2+d2 , then u ∈ D([0,T ];Cα1∨α2

loc (Rd1; Rd2)) and we can obtain arepresentation of the solution of (4.3.1) as in Theorem 3.2.2 by applying the Ito-Wentzellformula given in Proposition 3.4.16. However, there are two important points to notice.First, since we have only established an integer regularity theory for our equation, thenthe best we can hope for is a theory for classical solutions with integer assumptions onthe coefficients, initial condition, and free terms. This is in stark contrast with Chapter 3.Moreover, the restriction to p = 2 has dire consequences, since the d1

2 term gets larger asd1 grows, and hence one must impose more regularity as the dimension grows. This is astrong motivation to consider the Lp-Sobolev theory for (4.3.1) in weighted spaces, so thatthe d1

2 can be replaced with d1p . This way, by taking p large, the term d1

p can be made assmall as one likes. It is worth mentioning that we could have considered weighted scale ofSobolev spaces here (see, e.g. [GK92]), but we will leave this for a future project. At thetime of writing, the integer scale Lp-Sobolev theory for (degenerate) (4.3.1) is currentlyunderway and will be available soon.



By [MR99] (see Examples 2.3-2.4), the stochastic integrals in (4.3.1) can be written asstochastic integrals with respect to a cylindrical martingale. We will apply Theorem 4.2.4to (4.3.1) with α = m and µ = 1 by checking that Assumptions 4.2.1(λ, 1) and 4.2.2(λ, 1)for λ ∈ 0,m are implied by Assumption 4.3.1(m, d2). We start with λ = 0 as our basecase and show that λ = m can be reduced to it.

We introduce our base assumption for β ∈ [0, 2] .

Assumption 4.3.2 (d2). Let N0 be a positive constant.

(1) For all (ω, t, x) ∈ Ω × [0,T ] × Rd1 , the derivatives in x of the random fields bt, σ1t , σ

2t ,

and divσ1t exist, and for all x ∈ Rd1 ,

|∇ divσ1t (x) | + |σk

t (x) | + |∇σkt (x)| + |divbt(x)| + |ct(x)| + |υ2

t,sym (x) | + |∇υ1t (x) | ≤ N0.

(2) For each k ∈ 1, 2 and all (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × Zk, the derivatives in x ofthe random fields ζk

t (z) exist, and for all x ∈ Rd1 ,

|ζ1t (x, z)| ≤ K1

t (z) , |∇ζ1t (x, z) | ≤ K1

t (z) ,[div ζ1

t (·, z)]β2≤ K1

t (z) , ∀z ∈ Z1,

|ζ2t (x, z)| ≤ K2

t (z) , |∇ζ2t (x, z) | ≤ K2

t (z) , ∀z ∈ Z2,

where K1t , K

1t , K

1t (resp. K2

t , K2t ) are PT ⊗ Z

1 (resp. PT ⊗ Z2) -measurable processes

satisfying

supz∈Z1

(K1

t (z) + K1t (z) + K1

t (z))+

∫Z1

(K1

t (z)β + K1t (z)2 + K1

t (z)2)π1

t (dz) ≤ N0,

supz∈Z2

(K2

t (z) + K2t (z)

)+

∫Z2

K2t (z)2π2(dz) ≤ N0.

(3) There is a constant η < 1 such that for each k ∈ 1, 2 on the set all (ω, t, x, z) ∈ Ω ×[0,T ] × Rd1 × Zk for which |∇ζk

t (x, z)| > η,∣∣∣∣(Id1 + ∇ζkt (x, z)

)−1∣∣∣∣ ≤ N0.

(4) For each k ∈ 1, 2 and all (ω, t, x, z) ∈ Ω × [0,T ] × Rd1 × Zk, |ρkt (x, z)| ≤ lk

t (z), and forall (ω, t, z) ∈ Ω × [0,T ] × Z1,

[ρ1

t,sym(·, z)]β2≤ l1

t (z) , where lk (resp., l1) is a PT ⊗ Zk-

measurable (resp. PT ⊗Z1-measurable) functions satisfying∫

Z1

(l1t (z)2 + l1

t (z)2)π1(dz) +

∫Z2

l2t (z)2π2(dz) ≤ N0.


Note that Assumption 4.3.2(d2) is weaker than Assumption 4.3.1(0, d2).Let us make the following convention for the remainder of this section. If we do

not specify to which space the parameters ω, t, x, y, and z belong, then we mean ω ∈ Ω,t ∈ [0,T ], x ∈ Rd1 , and z ∈ Zk. Moreover, unless otherwise specified, all statements holdfor all ω, t, x, y, and z independent of any constant N introduced is independent of ω, t, x, y,and z. We will also drop the dependence of processes t, x, and z when we feel it will notobscure our argument. Lastly, all derivatives and Hölder norms are taken with respect tox ∈ Rd1 .

Remark 4.3.5. Let Assumption 4.3.2(d2) hold. For each k and θ ∈ [0, 1], on the set all ω, t,and z in which |Kk

t (z)| ≤ η, we have

∣∣∣(Id1 + θ∇ζkt (x, z))−1

∣∣∣ ≤ 11 − θη

.

Moreover, for each k and all ω, t, and z, we have

∣∣∣(Id1 + ∇ζkt (x, z))−1

∣∣∣ ≤ max(

11 − θη

,N0

).

Therefore, by Hadamard’s theorem (see, e.g., Theorem 0.2 in [DMGZ94] or 51.5 in[Ber77]):

• for each k and θ ∈ [0, 1], on the set all ω, t, and z in which |Kkt (z)| ≤ η, the mapping

ζkt,θ(x, z) := x + θζk

t (x, z)

is a global diffeomorphism in x;

• for each k and all ω, t, and z, the mapping

ζkt (x, z) = ζk

t,1(x, z) = x + ζkt (x, z)

is a global diffeomorphism in x.

When inverse of the mapping x 7→ ζkt,θ(x, z) exists, we denote it by

ζk;−1t,θ (x, z) =

(ζ

k;−1; jt,θ (x, z)

)1≤ j≤d1

and note thatζk;−1

t,θ (x, z) = x − θζkt (ζk;−1

t,θ (x, z), z).

Furthermore, for each k and θ ∈ [0, 1], on the set all ω, t, and z in which |Kkt (z)| ≤ η, there


is a constant N = N(d1,N0, η) such that

|∇ζk;−1t,θ (x, z)| ≤ N (4.3.2)

and for each k and all ω, t, and z,

|∇ζk;−1t (x, z)| ≤ N. (4.3.3)

Using simple properties of the determinant, we can easily show that there is a constantN = N(d1) such that for an arbitrary real-valued d1 × d1 matrix A,

| det(Id1 + A) − 1| ≤ N|A| and | det(Id1 + A) − 1 − tr A| ≤ N|A|2.

Thus, there is a constant N = N(d1,N0, η) such that

| det∇ζk;−1 − 1| =∣∣∣∣det

(Id − ζ

kt (ζk;−1

t ))− 1

∣∣∣∣ ≤ N |∇ζk(ζk;−1)| (4.3.4)

and ∣∣∣∣det∇ζk;−1 − 1 + div(ζk(ζk;−1)

)∣∣∣∣ ≤ N |∇ζk(ζk;−1)|2.

Since ∂lζk;−1; j = δl j − ∂mζ

k; j(ζk;−1)∂lζk;−1;m), we have

| div(ζk(ζk;−1)) − div ζk(ζk;−1)| = |∂ jζk;l(ζk;−1

t )(∂lζk;−1; j − δl j)| ≤ N|∇ζk(ζk;−1)|2,

and thus ∣∣∣det∇ζk;−1 − 1 + div ζk(ζk;−1)∣∣∣ ≤ N |∇ζk(ζk;−1)|2. (4.3.5)

In the following three lemmas, we will show that Assumptions 4.2.1(λ, 1) and 4.2.2(λ, 1)for λ ∈ 0,m hold under Assumption 4.3.2(β) for any β ∈ [0, 2]. For each l ∈ 1, . . . , d2

and all ϕ ∈ C∞c , let

Lltϕ = (L1;l

t +L2;lt )ϕ + bi

t∂iϕl + cll

t ϕl,

A1;lt ϕ =

12σ

1;iϱt σ

1; jϱt ∂i jϕ

l + σ1;iϱt υ

1;llϱt ∂iϕ

l, and J1t ϕ = L

1t ϕ −A

1t ϕ.

Lemma 4.3.6. Let Assumption 4.3.2(d2) hold. Then there is a constant N = N(d1, d2,N0, η)such that for all (ω, t) ∈ Ω × [0,T ] and v ∈ H1,

∥Ltv∥−1 ≤ N∥v∥1, ∥Atv∥−1 ≤ N∥v∥1, ∥J1t v∥−1 ≤ N∥v∥1,

∥Ntv∥0 ≤ N∥ϕ∥1, and∫

Z1∥It,zv∥20π

1t (dz) ≤ N∥v∥21.


Proof. First we will show that there is a constant N such that

(ψ,Ltϕ)0 ≤ N∥ψ∥1∥ϕ∥1, ∀ϕ ∈ C∞c .

Once this is established, we know that L extends to a linear operator from H1 to H−1

(still denoted by L) and ∥Ltv∥−1 ≤ N∥v∥1, for all v ∈ H1. Using Taylor’s formula and thedivergence theorem, we get that for all and all ϕ, ψ ∈ C∞c

(ψ,Lϕ)0 =

2∑k=1

((ψ,Lk

tϕ)0 + (∂iψ,Yk;it ϕ)0 + (ψ, bt∂iϕ)0 + (ψ, ctϕ)0

),

where for each k ∈ 1, 2, l ∈ 1, . . . , d2, and i ∈ 1, 2, . . . , d1,

Lk;lϕ : = −

∫Kk<η

∫ 1

0

(ϕl(ζk

θ ) − ϕl)∂iζ

k;idθπk(dz)

−

∫Kk<η

∫ 1

0θ∂ jϕ

l(ζkθ )∂iζ

k; jζk;idθπk(dz)

+

∫Zkρk;ll

(ϕl(ζk) − ϕl

)πk(dz) + σk;iϱυk;llϱ∂iϕ

l

+

∫Kk>η

(ϕl(ζk) − ϕl − ζk;i∂iϕ

l)πk(dz),

Yk;liϕ : = −

∫Kk<η

∫ 1

0

(ϕl(ζk

θ ) − ϕl)ζk;idθπk(dz) −

12∂i

(σk;iϱσk; jϱ

)∂ jϕ

l.

For the remainder of the proof, we make the convention that statements hold for all ϕ, ψ ∈C∞c and that all constants N are independent of ϕ. By Minkowski’s integral inequality andHölder’s inequality, we have (using the notation of Remark 4.3.5)

||Lkϕ||0 ≤

(∫Kk<η

(Kk)2πk(dz)) 1

2∫ 1

0

(∫Rd1

∫Kk<η

|ϕ(ζkθ (z)) − ϕ|2πk(dz)dx

) 12

dθ

+

∫Kk<η

Kk(z)2∫ 1

0

(∫Rd1

|∇ϕ(ζkθ )|

2dx) 1

2

πk(dz)θdθ

+

(∫Z1

(lk)2πk(dz)) 1

2(∫

Rd1

∫Z1|ϕ(ζk) − ϕ|2πk(dz)dx

) 12

+N∥∇ϕ∥0 +∫

Kk≥η

∫ 1

0

(∫Rd1

|ϕ(ζk) − ϕ − ζk;i∂iϕ|2dx

) 12

dθπk(dz),

and for all i ∈ 1, . . . , d1,

||Yk;iϕ||0 ≤

(∫Kk<η

(Kk)2πk(dz)) 1

2∫ 1

0

(∫Rd1

∫Kk<η

|ϕ(ζkθ ) − ϕ|

2πk(dz)dx) 1

2

dθ + N∥∇ϕ∥0.


Applying the change of variable formula and appealing to (4.3.2), we find that∫Rd1

∫Kk<η

|ϕ(ζkθ ) − ϕ|

2πk(dz)dx ≤ θ∫

Kk<η

∫ 1

0

∫Rd1

|∇ϕ(ζkθθ

)|2|ζk|2dxdθ

≤ θ

∫Kk<η

(Kk)2∫ 1

0

∫Rd1

|∇ϕ|2∣∣∣det∇ζk;−1

θθ

∣∣∣ dxdθ ≤ Nθ∥∇ϕ∥20.

Similarly, since πk(z ∈ Zk : Kk ≥ η) ≤ N0, we have∫Rd1

∫Kk≥η

|ϕ(ζk) − ϕ|2πk(dz)dx ≤ 2∫

Rd1

∫Kk≥η

(|ϕ(ζk)|2 + |ϕ|2

)πk(dz)dx

≤

∫Kk≥η

∫Rd1

|ϕ|2(1 +

∣∣∣det∇ζk;−1∣∣∣) dxπk(dz) ≤ N∥ϕ∥20

and ∫Kk≥η

(∫Rd1

|ϕ(ζk) − ϕ − ζk;i∂iϕ|2dx

) 12

πk(dz)

≤ N∫

Kk≥η

(∫Rd1

(|ϕ|2

(1 +

∣∣∣det∇ζk;−1∣∣∣) + (Kk)2|∇ϕ|2

)dx

) 12

πk(dz) ≤ N∥ϕ∥1,

where in the last inequality we used (4.3.3). Moreover,

∫Kk<η

(Kk)2(∫

Rd1

|∇ϕ(ζkθ )|

2dx) 1

2

πk(dz)

≤

∫Kk<η

(Kk)2(∫

Rd1

|∇ϕ|2 det∇ζk;−1θ dx

) 12

πk(dz) ≤ N∥∇ϕ∥0.

Combining the above estimates, we get that (ψ,Lϕ)0 ≤ N∥ψ∥1∥ϕ∥1. It is clear from theabove computation that

∥A1t ϕ∥−1 ≤ N∥ϕ∥1, ∥J1

t ϕ∥−1 ≤ N∥ϕ∥1,

where actually A1 and J1 are actually extensions of the operators defined above. Theinequality ∥Nϕ∥0 ≤ N∥ϕ∥1 can easily be obtained. Following similar calculations to oneswe derived above (using (4.3.3) and (4.3.2)), we obtain∫

Z1||Iϕ||20π

1(dz) ≤ N(A1 + A2),


where

A1 :=∫

Rd1

∫K1≤η

∫ 1

0|∇ϕ(ζ1

θ )|2|ξ1|2π1(dz)dθdx

+

∫Rd1

∫K1>η

(|ϕ(ζ1)|2 + |ϕ|2

)π1(dz)dx ≤ N∥ϕ∥21

andA2 :=

∫Rd1

∫Z1ρ1;llϕl(ζ1)π1 (dz) dx ≤ N∥ϕ∥20.

Lemma 4.3.7. Let Assumption 4.3.2(d2) hold. Then there is a constant N = N(d1, d2,N0, η, β)such that for all (ω, t) ∈ Ω × [0,T ] and all v ∈ H1,

2⟨v,L2t v + bi

t∂ivt + c·lt vlt⟩1 +

14

(σ2;iϱt ∂iv, σ

2; jϱt ∂ jv)0

+14

∫Z2∥v(ζ2

t (z)) − v∥20π2t (dz) ≤ N||v||20, (4.3.6)

2⟨v,A1t v⟩1 + ||Ntv||20 ≤ N||v||20, 2⟨v,J1

t v⟩1 +∫

Z1||It,zv||20π

1t (dz) ≤ N||v||20, (4.3.7)

and

2⟨v,Ltv + ft⟩1 + ||Ntv + gt||20 +

∫Z1

||It,zv + ht (z) ||20π1 (dz)

+14

(σ2;iϱt ∂iv, σ

2; jϱt ∂ jv)0 +

14

∫Z2∥v(ζ2

t (z)) − v∥20π2t (dz)

≤ N(∥v∥20 + || ft||

20 + ||gt||

21 +

∫Z1||ht (z) ||2β

2π1

t (dz)).

Proof. For the remainder of the proof, we make the convention that statements hold forall ϕ ∈ C∞c and that all constants N are independent of ϕ. Using the divergence theorem,we get

2⟨ϕ,A1ϕ⟩1 + ||Nϕ||20 =

12

∫Rd1

(| divσ1|2 + 2σ1;i∂i divσ1 + ∂ jσ

1;i∂iσ1; j

)|ϕ|2dx

+

∫Rd1

(|υ1ϕ|2 − 2ϕl

(υ1;ll

sym divσ1 + σ1;i∂iυ1;llsym

)ϕl

)dx ≤ N∥ϕ∥20.

Rearranging terms and using the identity 2a(b − a) = −|b − a|2 + |b|2 − |a|2, a, b ∈ R, weobtain

2⟨ϕ,J1ϕ⟩1 +

∫Z1

||Iϕ||20π1t (dz) = A1 + A2,


where

A1 :=∫

Rd1

∫Z1

(|ϕ(ζ1)|2 − |ϕ|2 − 2ϕζ1;i∂iϕ

)π1(dz)dx

A2 := 2∫

Rd1

∫Z1

(ϕl(ζ1)ρ1;llϕl(ζ1) − ϕlρ1;lϕl

)π1(dz)dx +

∫Rd1

∫Z1|ρ1ϕ(ζ1)|2π1(dz)dx,

Since| div ζ1(ζ1;−1) − div ζ1| ≤ [div ζ1] β

2(K1)

β2 ≤ (K1)2 + (K1)β,

changing the variable of integration and making use of the estimate (4.3.5), we obtain

A1 ≤

∫Rd1

|ϕ|2∫

Z1| det∇ζ1;−1 − 1 + div ζ1|π1(dz)dx ≤ N∥ϕ∥20

and

A2 = 2∫

Rd1

∫Z1ϕl

(ρ1;ll(ζ1;−1) − ρ1;ll

)ϕlπ1(dz)dx

+

∫Rd1

∫Z1

2ϕlρ1;ll(ζ1;−1)ϕl(det∇ζ1;−1 − 1

)π1(dz)dx

+

∫Rd1

∫Z1|ρ1(ζ1;−1)ϕ|2 det∇ζ1;−1π1(dz)dx =: A21 + A22 + A23.

Owing to (4.3.4) and Hölder’s inequality, we have

A22 + A23 ≤ N∫

Z1((l1)2 + (K1)2)π1(dz)||ϕ||20.

For β > 0, we have

A21 ≤ N∫

Z1

[ρ1

sym

]β2

(K1)β2π1(dz)∥ϕ∥20 ≤ N

∫Z1

((l1)2 + (K1)β

)π1(dz)∥ϕ∥20 ≤ N∥ϕ∥20

and for β = 0, using Holder’s inequality, we get

A21 ≤ N∥ϕ∥20

∫Z1

(l1)2π1(dz).

By the divergence theorem, we have

2⟨ϕ,L2ϕ⟩0 = B1 + B2 + B3,

where

B1 :=∫

Rd1

(ϕlσ2;iϱσ2; jϱ∂i jϕ

l + 2σ2;iϱυ2;llϱ∂iϕl)

dx,


B2 := 2∫

Rd1

∫Z2ϕl

(ϕl(ζ2) − ϕl − ζ2;i∂iϕ

l)π2(dz)dx,

B3 := 2∫

Rd1

∫Z2ϕlρ2;ll

(ϕl(ζ2) − ϕl

)π2(dz)dx.

Owing to the divergence theorem, we have

(ϕ, σ2;iϱσ2; jϱ∂i jϕ)0 = −((σ2;iϱσ2; jϱ∂iϕ + ϕ

(σ2; jϱ divσ2;ϱ + σ2;iϱ∂iσ

2; jϱ)), ∂ jϕ

)0

(ϕσ2;iϱ∂iσ2; jϱ, ∂ jϕ)0 = −

12

(ϕ(∂ jσ

2;iϱ∂iσ2; jϱ + σ2; jϱ∂ j divσ2;ϱ

), ϕ

)0,

(ϕσ2; jϱ∂ j divσ2;ϱ, ϕ)0 = −(ϕ| divσ2|2, ϕ)0 + 2(ϕσ2;iϱ divσ2;ϱ, ∂ jϕ)0,

and hence,

(ϕ, σ2;iϱt σ

2; jϱt ∂i jϕ)0 = −

(σ2;iϱσ2; jϱ∂iϕ

l∂ jϕl + 2ϕσ2; jϱ divσ2;ϱ, ∂ jϕ

)0

+12

(ϕ(∂ jσ

2;iϱ∂iσ2; jϱ − | divσ2|2

), ϕ

)0.

Thus, by Young’s inequality,

B1 ≤ −12

∫∂iϕ

lσ2;iϱσ2; jϱ∂ jϕldx + N∥ϕ∥20.

Once again making use of the identity 2a(b − a) = −|b − a|2 + |b|2 − |a|2, a, b ∈ R, we get

2ϕl(ϕl(ζ2) − ϕl − ζ2;i∂iϕl) = −|ϕ(ζ2) − ϕ|2 + |ϕ(ζ2)|2 − |ϕ|2 − ζ2;i

t ∂i|ϕ|2.

Changing the variable of integration and applying the divergence theorem, we obtain

B2 = −

∫Rd1

∫Z2|ϕ(ζ2) − ϕ|2π2(dz)dx

+

∫Rd1

∫Z2|ϕ|2

(det∇ζ2;−1 − 1 + div ζ2(ζ2;−1)

)π2(dz)dx

+

∫Rd1

∫Z2|ϕ|2

(div ζ2 − div ζ2(ζ2;−1)

)π2(dz)dx.

Changing the variable of integration in the last term of B2, we get∫Rd1

∫Z2|ϕ|2

(div ζ2 − div ζ2(ζ2;−1)

)π2(dz)dx

=

∫Rd1

∫Z2

(|ϕ|2 − |ϕ(ζ)|2 det∇ζ2

)div ζ2π2(dz)dx

=

∫Rd1

∫Z2|ϕ(ζ2)|2

(1 − det∇ζ2

)div ζ2π2(dz)dx


+

∫Rd1

∫Z2

(ϕl − ϕl(ζ2))(ϕl + ϕl(ζ2)) div ζ2π2(dz)dx =: B21 + B22.

Clearly,

B21 ≤ N∫

Z2(K2)2π2(dz)∥ϕ∥20,

and applying Hölder’s inequality,

B22 ≤ N∫

Rd1

(∫Z2|ϕ(ζ2) − ϕ|2π2(dz)

) 12(∫

Z2

(|ϕ|2 + |ϕ(ζ2)|2

)(K2)2π2(dz)

) 12

dx.

Hence, by Remark 4.3.5 and Young’s inequality,

B2 ≤ −12

∫Rd1

∫Z2|ϕ(ζ2) − ϕ|2π2(dz)dx + N∥ϕ∥20.

By Hölder’s inequality,

B3 ≤ N∫

Rd1

(∫Z2|ϕ(ζ2) − ϕ|2π2(dz)

) 12(∫

Z2(l2)2π2(dz)

) 12

|ϕ|dx.

Applying Young’s inequality again and combining B2 and B3, we derive

2⟨ϕ,L2ϕ⟩1 ≤ N∥ϕ∥20 −14

∫∂iϕ

lσ2;iϱσ2; jϱ∂ jϕldx −

14

∫ ∫Z2|ϕ(ζ2) − ϕ|2π2(dz)dx. (4.3.8)

By the divergence theorem, we have

2⟨ϕ, bi∂iϕ + c·lϕl + f ⟩0 = 2 (ϕ, f )0 + (ϕ, ϕ div b)0 + 2(ϕ, cϕ)0 ≤ N(∥ϕ∥20 + ∥ f ∥20). (4.3.9)

Combining (4.3.8) and (4.3.9), we obtain (4.3.6). To obtain the estimate (4.3.7), we use(4.3.6) and (4.3.7), and estimate the additional terms:

D :=(σ1;iϱ∂iϕ + υ

1;·lϱϕl, gϱ)

0

and2∫

Z1

((ϕ(ζ1) − ϕ, h

)0+ (ρ1ϕ(ζ1), h)0

)π1(dz) =: E1 + E2.

By the divergence theorem and Hölder’s inequality,|D| ≤ N(||ϕ||20 + ||g||

21

).Applying Hölder’s

inequality and changing the variable of integration, we get

E2 ≤

∫Rd1

∫Z1

(|ρ1ϕ(ζ1)|2 + |h|2

)π1(dz)dx ≤ N

(||ϕ||20 +

∫Z1||h (z) ||20π

1(dz)).


Then by (4.3.4), Hölder’s inequality, and Lemma 4.3.10,

E1 = 2∫

Z1

∫Rd1

ϕl(hl(ζ1;−1)(det∇ζ1;−1 − 1) + hl(ζ1;−1) − h

)dxπ1(dz)

≤ N(∥ϕ∥20 +

∫Rd1

∫Z1

(|h|2 + |h(ζ1;−1) − h|2

)π1(dz)dx

)≤ N

(∥ϕ∥20 +

∫Z1||h(z)||2β

2π1(dz)

).

This completes the proof.

In the following lemma, we verify that Assumption 4.2.2(0, 1) holds for (4.3.1). RecallthatW0,1 is the space of all H0-valued strongly càdlàg processes v : Ω × [0,T ]→ H0 thatbelong to L2(Ω × [0,T ],OT , dVtdP; H1).

Lemma 4.3.8. Let Assumption 4.3.2(d2) hold. Then there is a constant N = N(d1, d2,N0, η, β)such that for all v ∈ W0,1, P-a.s.:

(1)

2⟨vt,Ltvt⟩1dVt + ∥Ntvt∥20dVt +

∫Z1∥It,zvt−∥

20η(dt, dz)

+2(vt,Nϱt vt)0dwϱ

t + 2∫

Z1(vt−,It,zvt−)0η (dt, dz)

≤

(N||vt||

20dVt +

∫Z1

Nκt(z)||vt−||20η(dt, dz) + 2(vt,N

ρt vt)0dwϱ

t +

∫Z1

Gt,z(v)η(dt, dz)),

where

|Gt,z(v)|dVt ≤ κt(z)||vt−||20dVt, ∀z ∈ Z1, |(vt,Ntvt)0|dVt ≤ N ||vt||

20 dVt,

and κt and κt are PT ×Z1-measurable processes such that for all t ∈ [0,T ],∫

Z1

(κt(z) + κt(z)2

)π1

t (dz) ≤ N;

(2)

2(Nϱt vt, g

ϱt )0dVt + 2

∫Z1

(It,zvt−, ht(z))0η(dt, dz) + 2(vt, gϱt )0dwϱ

t + 2∫

Z1(vt−, ht(z))0η(dt, dz)

≤

(N∥vt−∥0rtdVt +

∫Z1

N∥vt−∥0∥ht(z)∥0κt(z)η(dt, dz) + 2(vt, gϱt )0dwϱ

t + 2∫

Z1Gt,z(v)η(dt, dz)

),


where

rt := ∥gt∥1 +

∣∣∣∣∣∣∣∣∣∣∫Z1

(ht(ζ1;−1

t (z), z) − ht(z))π1

t (dz)∣∣∣∣∣∣∣∣∣∣

0, t ∈ [0,T ],

|(vt, gt)0|dVt ≤ N∥vt∥0∥gt∥0dVt,

Gt,z(v)dVt ≤ N ||vt−||0||ht(z)||0, dVt, ∀z ∈ Z1,

and κt is a PT ×Z1-measurable process such that for all t ∈ [0,T ],∫

Z1κt(z)2π1

t (dz) ≤ N.

Proof. (1) Owing to the divergence theorem, we have

2(vt,Nϱt vt)0 = (vt, ut divσ1ϱ

t )0 + 2(vt, υ1ϱt vt)0, ∀ϱ ∈ N,

and hence P-a.s.,|2(vt,Ntvt)0|dVt ≤ N||vt||

2dVt.

By virtue of Lemma 4.3.7(1), it suffices to estimate

Q := 2⟨vt,J1t,zvt⟩1dVt +

∫Z1∥It,zvt−∥

20η(dt, dz) + 2

∫Z1

(vt−,It,zvt−)0η (dt, dz) .

An application of divergence theorem shows that

Q =∫

Z1Pt,z(u)η(dt, dz) +

∫Z1

Gt,z(v)η(dt, dz),

whereGt,z(v) := 2(vt−, ρ

1t (z)vt−)0 − (vt−, vt− div ζ1

t (z))0,

and Pt,z(v) := D1 + D2 + D3 with

D1 := 2(vt−(ζ1t (z)), ρ1

t (z)vt−(ζ1t ))0 − 2(vt−, ρ

1t (z)vt−)0,

D2 := ∥vt−(ζ1t (z))∥20 − ∥vt−∥

20 + (vt−, vt− div ζ1

t (z))0, D3 := ∥ρ1t (z)vt−(ζ1

t (z))∥20.

Given our assumptions, it is clear that P-a.s.,

Gt,z(v)dVt ≤ N(l1t (z) + K1

t (z))∥vt−∥

20dVt and D3dVt ≤ l1

t (z)2||vt−||

20dVt,

where in the last inequality we used the change of variable formula. Changing the variable


of integration and using (4.3.4) and (4.3.5), we find that dP -a.s.,

D1dVt ≤ N(vt−, vt−

∣∣∣ρ1t (ζ1;−1

t (z), z) det∇ζ1;−1t (z) − ρ1

t (z)∣∣∣)

0dVt

≤ N(l1t (z)K1

t (z) + l1t (z)K1

t (z)β2

)∥vt−∥

20dVt,

and

D2dVt =(vt−, vt−| det∇ζ1;−1

t (z) − 1 + div ζ1t (z)|

)0

dVt ≤

(K1

t (z)2 + K1t (z)K1

t (z)β2

)N∥vt−∥

20dVt.

Setting

κt(z) = l1t (z)2+ l1

t K1t (z)+ l1

t (z)K1t (z)

β2 + K1

t (z)2+ K1t (z)K1

t (z)β2 , κt(z) = l1

t (z)+ K1t (z), z ∈ Z1,

and appealing to our assumptions, we complete the proof (1).(2) By the divergence theorem, we have

(gϱt ,Nϱt vt)0 = (gϱt , divσ1ϱ

t vt)0 + (σ1;iϱ∂igϱt , vt)0, ∀ρ ∈ N,

and thus by the Cauchy-Schwarz inequality,

|(gt,Ntvt)0|dVt ≤ N∥vt∥0∥gt∥1dVt.

Changing the variable of integration, we obtain

(It,zvt−, ht(z))0 =(ht(z),

(vt−(ζ1

t (z)) − vt− + ρ1t (z)vt−(ζ1

t (z))))

0

= (ht(ζ1;−1t (z), z) − ht(z), vt−)0 + (ht(ζ1;−1

t (z), z), (det∇ζ1;−1t (z) − 1)vt−)0

+ (ht(z), ρ1t (z)vt−(ζ1

t (z)))0.

A simple calculation shows that P-a.s.,

2∫

Z1(It,zvt−, ht(z))0η(dt, dz) + 2

∫Z1

(vt−, ht(z))0η(dt, dz)

≤ 2∥vt−∥0r1t dVt +

∫Z1

Pt,z(v)η(dt, dz) +∫

Z1Gt,z(v)η(dt, dz),

where

r1t :=

∣∣∣∣∣∣∣∣∣∣∫Z1

(ht(ζ1;−1t (z), z) − ht(z))π1

t (dz)∣∣∣∣∣∣∣∣∣∣

0, Gt,z(v) = (ht(ζ1;−1

t (z), z), vt),

andPt,z(v) := (ht(ζ1;−1

t (z)), vt−(det∇ζ1;−1t (z) − 1))0 + (ht(z), ρ1

t (z)vt−(ζ1t (z)))0.


Applying the change of variable formula and Hölder’s inequality, P-a.s. we obtain∫Z1

Pt,z(u)η(dt, dz) ≤ N∥vt−∥0

∫Z1

(K1t (z) + l1

t (z))∥ht(z)∥0η(dt, dz)

and

|Gt,z(v)|dVt ≤ N∥vt−∥0∥ht(z)∥0dVt.


Let d ∈ N. For a function v ∈ Hm(Rd1 ,Rd), define the linear operator Dv ∈ Hm−1(Rd1;Rd(d1+1)) by

Dv =(∂0v, ∂1v, . . . , ∂d1v

)= v

with vl0 = vl and vl j = ∂ jvl, 1 ≤ l ≤ d, 0 ≤ j ≤ d1 (recall ∂0v = v). We define Dnv forn ∈ N by iteratively applyingD n-times. Recall that Λ = (I − ∆)

12 . It is easy to check that

for each n ∈ N and all u, v ∈ Hn+1(Rd1 ,Rd),

(u, v)n,d = (Λnu,Λnv)0,d = (Dnu,Dnv)0,ddn1, (4.3.10)

(Λu,Λ−1v)n,d = (Λn+1u,Λn−1v)0,d = (DnΛu,DnΛ−1v)0,ddn1,

(Dnu,Dnv)−1,ddn1= (DnΛ−1u,DnΛ−1v)0,ddn

1= (u, v)n−1,d,

where d1 = d1 + 1. Let us introduce the operators E(L), E(N), and E(Iz) acting on ϕ =(ϕl j)1≤l≤d2,1≤ j≤d1 ∈ C∞c (Rd1 ,Rd2d1) that are defined as L,N , and I, respectively, but withthe d2 × d2-dimensional coefficients υk

t , ρk, and c replaced by the d2d1 × d2d1-dimensional

coefficients given by

υk;l j,l jϱ = υk;llϱδ j j + 1 j≥1(∂ jσk; jϱδll + ∂ jυ

k;llϱδ j0),

ρk;l j,l j = ρk;llt δ j j + 1 j≥1(∂ jρ

k;llδ j0 + (δll + ρk;ll)∂ jζ

k; j),

and

cl j,l j = cllδ j j + ∂ jb jδll + ∂ jcllδ j0 +

2∑k=1

(υk;llϱ∂ jσ

k; jϱ +

∫Zkρk;ll∂ jζ

k; jπkt (dz)

),

for 1 ≤ l, l ≤ d2 and 0 ≤ j, j ≤ d1. The coefficients σk, b, and functions ζk, k ∈ 1, 2,remain unchanged in the definition of E(L), E(N), and E(I). We define En(L), En(N),and En(I), for n ∈ N by iteratively applying E n-times by the rules above with σk, b, andζk, k ∈ 1, 2, unchanged. A simple calculation shows that for all v ∈ H2(Rd1; Rd2),

D[Lv] = E(L)Dv, D[Nϱv] = E(Nϱ)Dv, ϱ ∈ N, D[Izv] = E(Iz)Dv.


Continuing, for all v ∈ Hn+1(Rd1; Rd) we have

Dn[Lv] = En(L)Dnv, Dn[Nϱv] = En(Nϱ)Dnv, ϱ ∈ N, Dn[Izv] = En(Iz)Dnv.(4.3.11)

If Assumption 4.3.1(m, d2) holds, it can readily be verified by induction and the definitions(4.3.2)-(4.3.2) that Assumption 4.3.2(0, d2dm

1 ) holds for the coefficients of the operatorsEm(L), Em(N), and Em(I). Moreover, owing to our assumptions on the input data, wehave

Dmϕ ∈ H0(Rd1; Rd2dm1 ;F0), Dm f ∈ H0(Rd1; Rd2dm

1 )

Dmg ∈ ζ1(Rd1; ℓ2(Rd2dm1 )), Dmh ∈ H

β2 (Rd1; Rd2dm

1 ; π1).

Making use of (4.3.10), (4.3.11) and applying Lemma 4.3.6 to Em(L), for all v ∈ Hm+1, weobtain

∥Lv∥m−1 = ∥Dm[Lv]∥−1 = ∥E

m(L)Dmv∥−1,d2dm1≤ N∥Dmv∥1,d2dm

1= N∥v∥m+1.

Likewise, for all v ∈ Hm+1, we derive

∥Nv∥m ≤ N∥v∥m+1,

∫Z1||Iv||2mπ

1(dz) ≤ N∥v∥2m+1.

By virtue of Lemma 4.3.7, we have that for all v ∈ Hm+1,

2(Λv,Λ−1Ltv)m + ||Ntv||2m +∫

Z1||It,zv||2mπ

1 (dz)

= 2(DmΛv,DmΛ−1[Ltv])0 + ||Dn[Ntv]||20 +

∫Z1||Dm[It,zv]||20π

1(dz)

2⟨Dmv,Em(Lt)Dmv⟩1 + ||Em(Nt)Dmv||20 +∫

Z1||Em(It,z)Dnv||20π

1(dz) ≤ N ||Dmv||20,d2dm1= N ||v||2m

Using a similar argument, we find that for all v ∈ Hm+1,

2(Λv,Λ−1(Ltv + ft))m + ||Ntv + gt||2m +

∫Z1||It,zv + ht(z)||2mπ

1(dz) ≤ N||v||2m + N ft,

whereft = ∥ ft∥

2m + ∥gt∥

2m+1 +

∫Z1∥ht(z)∥2

m+ β2π1

t (dz).

Therefore, Assumption 4.2.1(m, d2) holds for the equation (4.3.1). Similarly, using Lem-mas 4.3.8 and 4.3.10, we find that Assumption 4.2.2(m, d2) holds for equation (4.3.1) aswell. The statement of the theorem then follows directly from Theorem 4.2.4.


4.3.3 Appendix

For each κ ∈ (0, 1) and tempered distribution f on Rd1 , we define

∂κ f = F −1[| · |κF f (·)],

where F denotes the Fourier transform and F −1 denotes the inverse Fourier transform.

Lemma 4.3.9 (cf. Lemma 2.1 in [Kom84]). Let f : Rd1 → R be smooth and bounded.Then for all κ ∈ (0, 1), there are constants N1 = N1(d1, κ), N2 = N2(d1, κ), and N3 =

N2(d1, κ) such that for all x, y, z ∈ Rd1 ,

∂κ f (x) = N1

∫Rd

( f (x + z) − f (x))dz|z|d+δ

andf (x + y) − f (x) = N2

∫Rd1

∂κ f (x − z)k(κ)(y, z)dz,

wherek(κ)(y, z) = |y + z|κ−d − |z|κ−d and

∫Rd1

|k(κ)(y, z)|dz = N3|z|κ.

Lemma 4.3.10. Let (Z,Z, π) be a sigma-finite measure space. Let H : Rd1 × Z → Rd1 beB(Rd) ⊗Z-measurable and assume that for all (x, z) ∈ Rd1 × Z,

|ζ(x, z)| ≤ K(z) and |∇ζ(x, z)| ≤ K(z)

where K, K : Z → R+ is a Z-measurable function for which there is a positive constantN0 such that for some fixed β ∈ (0, 2],

supz∈Z

K(z) + supz∈Z

K(z) +∫

Z

(K(z)β + K(z)2

)π(dz) < N0

Assume that there is a constant η < 1 such that (x, z) ∈ (x, z) ∈ Rd1 × Z : |∇ζ(x, z)| > η,

|(Id1 + ∇ζt(x, z)

)−1| ≤ N0.

Then there is a constant N = N(d1,N0, β, η) such that for all B(Rd1) ⊗ Z-measurableh : Rd1 × Z → Rd2 with h ∈ L2(Z,Z, π; H

β2 (Rd1; Rd2)),∫

Rd1

∣∣∣∣∣∫Z

(h(x + ζ(x, z), z) − h(x, z)) π(dz)∣∣∣∣∣2 dx ≤ N

∫Z∥h(z)∥2β

2π(dz).


Proof. It is easy to see that for any B(Rd1) ⊗Z-measurable h : Rd1 × Z → Rd2 such that∫Z

supx∈Rd1

|∇h(x, z)|2π(dz) < ∞, (4.3.12)

the integral∫

Z(h(x+ζ(x, z), z)−h(x, z))π(dz) is well-defined. Moreover, for anyB(Rd1)⊗Z-

measurable h : Rd1 × Z → R with h ∈ L2(Z,Z, π; Hβ2 (Rd1; Rd2)), we can always find

a sequence (hn)n∈N of B(Rd1) ⊗ Z-measurable processes such that each element of thesequence is smooth with compact support in x and satisfies (4.3.12) and

limn→∞

∫Z∥h(z) − hn(z)∥2β

2π(dz) = 0.

Thus, if we prove this lemma for h that is smooth with compact support in x and satisfies(4.3.12), then we can conclude that the sequence∫

Z(hn(x + ζ(x, z), z) − hn(x, z)) π(dz), n ∈ N,

is Cauchy in H0(Rd1; Rd2). We then define∫Z

(h(x + ζ(x, z), z) − h(x, z)) π(dz)

for any B(Rd1) ⊗ Z-measurable h : Rd1 × Z → R with h ∈ L2(Z,Z, π; Hβ2 (Rd1; Rd2)) to be

the unique H0(Rd1; Rd2) limit of the Cauchy sequence. Hence, it suffices to consider h thatis smooth with compact support in x and satisfies (4.3.12). First, let us consider the caseβ ∈ (0, 2). By Lemma 4.3.9, we have∫

Rd1

∣∣∣∣∣∫Z

(h(ζ(x, z), z) − h(x, z)

)π(dz)

∣∣∣∣∣2 dx

= N22

∫Rd1

∣∣∣∣∣∫Z

∫Rd1

∂β2 h(x − y, z)k( β2 )(ζ(x, z), y) dyπ(dz)

∣∣∣∣∣2 dx

=: N22

∫Rd1

|

∫Z

A(x, z)π(dz)|2dx.

Applying Hölder’s inequality and Lemma 4.3.9, for all x and z, we have

A(x, z) ≤(∫

Rd1

|∂β/2h(x − y, z)|2k( β2 )(ζ(x, z), y)) dy) 1

2(∫

Rd1

k( β2 )(ζ(x, z), y)) dy) 1

2

=√

N3

(∫Rd1

|∂β/2h(x − y, z)|2k( β2 )(ζ(x, z), y) dy) 1

2

|ζt(x, z)|β4


≤ K(z)β2√

N3

(∫Rd1

|∂β2 h(x − y, z)|2k( β2 )(ζ(x, z), y) dyK(z)−

β2

) 12

.

Using Hölder’s inequality again, for all x, we get∣∣∣∣∣∫Z

A(x, z)π(dz)∣∣∣∣∣2 ≤ N3N0

∫Z

∫Rd1

|∂β/2h(x − y, z)|2k( β2 )(ζ(x, z), y) dyK(z)−β2π(dz).

For each x and z, we set

B(x, z) =∫|y|≤2K(z)

|∂β2 h(x − y, z)|2k( β2 )(ζ(x, z), y) dy

C(x, z) =∫|y|>2K(z)

|∂β2 h(x − y, z)|2k( β2 )(ζ(x, z), y) dy.

Changing the variable integration, for all x and z, we find

B(x, z) ≤∫|y+ζ(x,z)|≤3K(z)

|∂β2 h(x − y, z)|2

dy

|y + ζ(x, z)|d1−β2

+

∫|y|≤2K(z)

|∂β2 h(x − y, z)|2

dy

|y|d1−β2

=: B1(x, z) + B2(x, z),

and

B1(x, z) ≤∫|y|≤3K(z)

|∂β2 h((ζ(x, z) − y), z)|2|

dy

|y|d1−β2

≤ K(z)β2

∫|y|≤3|∂

β2 h((ζ(x, z) − yK(z)), z)|2

dy

|y|d1−β2

,

B2(x, z) ≤ K(z)β2

∫|y|≤2|∂β/2h(x − yK(z), z)|2

dy

|y|d1−β2

.

Owing to Remark 4.3.5, for all z, the map x 7→ x + ζ(x, z) = ζ(x, z) is a global diffeomor-phism and

det∇ζ−1(x, z) ≤ N.

for some constant N = N(N0, d1, η). Thus, by the change of variable formula, there is aconstant N = N(d1,N0, β, η) such that∫

Rd1

∫Z

B1(x, z)K(z)−β2π(dz)dx

≤

∫Z

∫|y|≤3

∫Rd1

|∂β/2h((ζ(x, z) − yK(z)), z)|2dxdy

|y|d1−β2

π(dz)

≤

∫Z

∫|y|≤3

∫Rd1

|∂β/2h((x − yK(z)), z)|2| det∇ζ−1(x, z)|dxdy

|y|d1−β2

π(dz)


≤ N∫

Z

∫Rd1

|∂β2 h(x, z)|2dxπ(dz)

and ∫Rd1

∫Z

K(z)β2 B2(x, z)dxπ(dz) ≤ N

∫Z

∫Rd1

|∂β2 h(x, z)|2dxπ(dz).

For all x, y, and z such that |ζ(x, z)| ≤ K(z) ≤ 12 |y|, we have∣∣∣∣∣∣∣ 1

|y + ζ(x, z)|d1−β2

−1

|y|d1−β/2

∣∣∣∣∣∣∣≤

∣∣∣∣∣d1 −β

2

∣∣∣∣∣∣∣∣∣∣∣∣ 1

|y + ζ(x, z)|1+d1−β2

+1

|y|1+d1−β2

∣∣∣∣∣∣∣ |ζ(x, z)| ≤ 3

∣∣∣∣∣d1 −β

2

∣∣∣∣∣ |ζ(x, z)|

|y|1+d1−β2

,

and hence for all x and z,

C(x, z) =∫|y|>2K(z)

|∂β2 h(x − y, z)|2

∣∣∣∣∣∣∣ 1

|y + ζ(x, z)|d−β2

−1

|y|d−β2

∣∣∣∣∣∣∣ dy

≤ N∫|y|>2K(z)

|∂β2 h(x − y, z)|2

|K(z)|

|y|1+d1−β2

dy

≤ NK(z)β2

∫|y|>2|∂

β2 h((x − K(z)y), z)|2

dy

|y|1+d1−β2

.

Estimating as above, we find that there is a constant N = N(d1,N0, β)∫Z

∫Rd1

K(z)β2 C(x, z)dxπ(dz) ≤ N

∫Z

∫Rd1

|∂β2 h(x, z)|2dxπ(dz).

Combining the above estimates, we obtain the desired estimate for β ∈ (0, 2). Let us nowconsider the case β = 2. It follows from Remark 4.3.5 that for all θ ∈ [0, 1], on the set ofz ∈ z : K(z) < 1

2 , the map x 7→ x + θζ(x, z) = ζθ(x, z) is a global diffeomorphism and

det∇ζ−1θ (x, z) ≤ N,

for some constant N = N(N0, d1). Hence, making use of Taylor’s theorem and the changeof variable formula, we find∫

Rd1

∣∣∣∣∣∫Z

(h(x + ζ(x, z), z) − h(x, z)) π(dz)∣∣∣∣∣2 dx

≤

∫Rd1

∣∣∣∣∣∣∫

K(z)≥ 12

(h(x + ζ(x, z), z) − h(x, z)) π(dz)

∣∣∣∣∣∣2 dx

+

∫Rd1

∣∣∣∣∣∣∫

K(z)< 12

∫ 1

0|∇h(x + θζ(x, z), z)

∣∣∣∣∣∣ dθK(z)π(dz)|2dx


≤ π

K(z) ≥

12

∫K(z)≥η

∫Rd1

|h(x, z)|2| det ζ−1(x, z) + 1|dxπ(dz)

+N0

∫K(z)< 1

2

∫Rd1

∫ 1

0|∇h(x, z)|2| det∇ζ−1

θ (x, z)|dθdxπ(dz) ≤ N∫

Z∥h(z)∥21π(dz).


Chapter 5A finite difference scheme fornon-degenerate parabolic SIDEs

5.1 Introduction

Let (Ω,F ,F,P), F = (Ft)t≥0, be a complete filtered probability space such that the filtrationis right continuous and F0 contains all P-null sets of F . Let wϱ

t , t ≥ 0, ϱ ∈ N, be a sequenceof independent real-valued F-adapted Wiener processes. Let π1(dz) and π2(dz) be a Borelsigma-finite measures on Rd satisfying∫

Rd|z|2 ∧ 1 πr(dz) < ∞, r ∈ 1, 2.

Let q(dt, dz) = p(dt, dz) − π2(dz)dt be a compensated F-adapted Poisson random measureon R+ × Rd. Let T > 0 be an arbitrary fixed constant. On [0,T ] × Rd, we consider finitedifference approximations for the following SIDE

dut = ((Lt + I)ut + ft) dt +∞∑ϱ=1

(N

ϱt ut + gϱt

)dwϱ

t +

∫Rd

(I(z)ut− + ot(z)) q(dt, dz),(5.1.1)

with initial conditionu0(x) = φ(x), x ∈ Rd,

where the operators are given by

Ltϕ(x) :=d∑

i, j=0

ai jt (x)∂i jϕ(x),

Iϕ(x) :=∫

Rd

ϕ(x + z) − ϕ(x) − 1[−1,1](|z|)d∑

j=1

z j∂ jϕ(x)

π1(dz), (5.1.2)

Nϱt ϕ(x) :=

d∑i=0

σiϱt (x)∂iϕ(x), I(z)ϕ(x) = ϕ(x + z) − ϕ(x).

Here, we denote the identity operator by ∂0.

128


Equation (5.1.1) arises naturally in non-linear filtering of jump-diffusion processes.We refer the reader to [Gri82] and [GM11] for more information about non-linear filter-ing of jump-diffusions and the derivation of the Zakai equation. Various methods havebeen developed to solve stochastic partial differential equations (SPDEs) numerically. ForSPDEs driven by continuous martingale noise see, for example, [GK96, Gyö98, Gyö99,GM09, LR04, JK10, Yan05], and for SPDEs driven by discontinuous martingale noise, see[HM06, Hau08, Lan12, BL12]. Among the various methods considered in the literatureis the method of finite differences. For second order linear SPDEs driven by continu-ous martingale noise it is well-known that the Lp(Ω)-pointwise error of approximation inspace is proportional to the parameter h of the finite difference (see, e.g., [Yoo00]). In[GM09], I. Gyöngy and A. Millet consider abstract discretization schemes for stochasticevolution equations driven by continuous martingale noise in the variational frameworkand, as a particular example, show that the L2(Ω)-pointwise rate of convergence of anEuler-Maruyuma (explicit and implicit) finite difference scheme is of order one in spaceand one-half in time. More recently, it was shown by I. Gyöngy and N.V. Krylov that undercertain regularity conditions, the rate of convergence in space of a semi-discretized finitedifference approximation of a linear second order SPDE driven by continuous martingalenoise can be accelerated to any order by Richardson’s extrapolation method. For the non-degenerate case, we refer to [GK10] and [GK11], and for the degenerate case, we referto [Gyö11]. In [Hal12] and [Hal13], E. Hall proved that the same method of accelerationcan be applied to implicit time-discretized SPDEs driven by continuous martingale noise.Also, for a pathwise convergence result of a spectral scheme for SPDEs on a boundeddomain we refer the reader to [CJM92].

In the literature, finite element, spectral, and, more generally, Galerkin schemes havebeen studied for SPDEs driven by discontinuous martingale noise. One of the earliestworks in this direction is a paper [HM06] by E. Hausenblas and I. Marchis concerningLp(Ω)-convergence of Galerkin approximation schemes for abstract stochastic evolutionequations in Banach spaces driven by Poisson noise of impulsive-type. As an applica-tion of their result, they study a spectral approximation of a linear SPDE in L2([0, 1])with Neumann boundary conditions driven by Poisson noise of impulsive-type and deriveLp(Ω)-error estimates in the L2([0, 1])-norm. In [Hau08], E. Hausenblas considers finiteelement approximations of linear SPDEs in polyhedral domains D driven by Poisson noiseof impulsive-type and derives Lp(Ω) error estimates in the Lp(D)-norm. In a more recentwork [Lan12], A. Lang studied semi-discrete Galerkin approximation schemes for SPDEsof advection diffusion type in bounded domains D driven by cádlág square integrable mar-tingales in a Hilbert Space. A. Lang showed that the rate of convergence in the Lp(Ω) andalmost-sure sense in the L2(D)-norm is of order two for a finite-element Galerkin scheme.In [BL12], A. Lang and A. Barth derive L2(Ω) and almost-sure estimates in the L2(D)-


norm for the error of a Milstein-Galerkin approximation scheme for the same equationconsidered in [Lan12] and obtain convergence of order two in space and order one in time.

In the articles [Lan12, BL12, HM06, Hau08], the authors make use of the semigrouptheory of SPDEs (mild solution) and only consider SPDEs in which the principal part ofthe operator in the drift is non-random. Moreover, the authors there do not address theapproximation of equations with non-local operators in the drift or noise. The principalpart of the operator in the drift of the Zakai equation is, in general, random, and hencenumerical schemes that approximate SPDEs or SIDEs with random-adapted principal partare of importance. More precisely, the coefficients of the Zakai equation are random if thecoefficients of the SDE governing the signal depend on the observation. In this chapter,since we use the variational framework (L2-theory) of SPDEs, we are easily able to treatthe case of random-coefficients, and hence the diffusion coefficients ai j

t (x) appearing in(5.1.1) are random.

In dimension one, a finite difference scheme for degenerate integro-differential equa-tions (deterministic) has been studied by R. Cont and E. Voltchkova in [CV05]. Theauthors in [CV05] first approximate the integral operator near the origin with a secondderivative operator. The resulting PDE is then non-degenerate and has an integral operatorof order zero. The error of this approximation is obtained by means of the probabilisticrepresentation of the solution of both the original equation and the non-degenerate equa-tion. In the second step of their approximation, R. Cont and E. Voltchkova consider animplicit-explicit finite difference scheme and obtain pointwise error estimates of order onein space. As a consequence of the two-step approximation scheme, there are two separateerrors for the approximation

In this chapter, we consider the non-degenerate stochastic integro-differential equation(5.1.1) with random coefficients and apply the method of finite differences in the timeand space variables. To the best of our knowledge, this article is the first to use the finitedifference method to approximate SIDEs. The approximations of the non-local integraloperators in the drift and in the noise of (5.1.1) we choose are both natural and relativelyeasy to implement. In particular, we are able to treat the singularity of the integral oper-ators near the origin directly. We consider a fully-explicit time-discretization scheme andan implicit-explicit time-discretization scheme, where we treat part of the approximationof the integral operator in the drift explicitly.

To obtain error estimates for our approximations, we use the approach in [Yoo00],where the discretized equations are first solved as time-discretized SDEs in Sobolev spacesover Rd and an error estimate is obtained in Sobolev norms. After obtaining L2(Ω) errorestimates in Sobolev norms, the Sobolev embedding theorem is used to obtain L2(Ω)-pointwise error estimates. So, in sum, we obtain two types of error estimates: in Sobolevnorms and on the grid. Naturally, when using the Sobolev embedding to obtain the point-


wise estimates, we do not need the equation to be differentiable to obtain pointwise errorestimates, only continuous. Using the approach of first obtaining estimates in Sobolevspaces, we are also easily able to deduce that the more regularity on the coefficients anddata we have, the stronger the error estimates we can obtain (see Corollaries 5.5.3 and5.5.4).

The chapter is organized as follows. In the next section, we introduce the notation thatwill be used throughout the chapter and state the main results. In the third section, wepresent a simulation that we did to confirm our rates of convergence. In the fourth section,we prove auxiliary results that will be used in the proof of the main theorems. In the fifthsection, we prove the main theorems of the chapter.


Let us consider the following assumption for an integer m ≥ 0.

Assumption 5.2.1 (m). For i, j ∈ 0, . . . , d, ai jt = ai j

t (x) are real-valued functions definedon Ω × [0,T ] × Rd that are RT ⊗ B(Rd)-measurable and σi

t = (σiϱt (x))∞ϱ=1 are ℓ2-valued

functions that are RT ⊗ B(Rd)-measurable. Moreover,

(i) for each (ω, t) ∈ Ω× [0,T ], the functions ai jt are max(m, 1)-times continuously differ-

entiable in x for all i, j ∈ 1, . . . , d, ai0t and a0i

t are m-time continuously differentiablein x for all i ∈ 0, 1, . . . , d, and σi

t are m-times continuously differentiable in x asℓ2-valued functions for all i ∈ 0, . . . , d. Furthermore, there is a constant K > 0such that for all (ω, t, x) ∈ Ω × [0,T ] × Rd,

|∂γai jt | ≤ K, ∀ i, j ∈ 1, . . . , d, ∀ |γ| ≤ max(m, 1),

|∂γai0t | + |∂

γa0it | + |∂

γσit|ℓ2 ≤ K, ∀ i ∈ 0, . . . , d, ∀|γ| ≤ m;

(ii) there exists a positive constant κ > 0 such that for all (ω, t, x) ∈ Ω × [0,T ] × Rd andη ∈ Rd

d∑i, j=1

2ai jt −

∞∑ϱ=1

σiϱt σ

jϱt

ηiη j ≥ κ|η|2.

In this chapter, for each integer m ≥ 0, we set Hm = Hm(Rd; R), Hm(F0), Hm =

Hm(Rd; R), Hm(ℓ2) = Hm(Rd; ℓ2), Hm(π2) = Hα(Rd; R; π2), and ∥ · ∥m = ∥ · ∥m,1, (·, ·)m =

(·, ·)m,1, ⟨·, ·⟩1 = ⟨·, ·⟩1,1. We also set C∞c = C∞c (Rd; R.

Assumption 5.2.2 (m). We have ϕ ∈ Hm(F0), f ∈ Hm−1, g ∈ Hm(ℓ2), and o ∈ Hm(π2). Set

κ2m = E

[∥φ∥2m

]+ E

∫]0,T ]

(∥ ft∥

2m−1 + ∥gt∥

2m,ℓ2+

∫Rd∥ot(z)∥2mπ

2(dz))

dt.


For a real-valued twice continuous differentiable function ϕ on Rd, it is easy to see thatfor all x, z ∈ Rd,

ϕ(x + z) − ϕ(x) −d∑

j=1

z j∂ jϕ(x) =∫ 1

0

d∑i, j=1

ziz j∂i jϕ(x + θz)(1 − θ)dθ. (5.2.1)

For each δ ∈ (0, 1], let

ς1(δ) =∫|z|≤δ|z|2π1(dz), ς2(δ) =

∫|z|≤δ|z|2π2(dz), and ς(δ) = ς1(δ) + ς2(δ).

Fix δ ∈ (0, 1] such thatς(δ) < κ, (5.2.2)

and notice that2∑

r=1

πr(|z| > δ) < ∞. (5.2.3)

We write I = Iδ + Iδc , where

Iδϕ(x) :=∫|z|≤δ

∫ 1

0

d∑i, j=1

ziz j∂i jϕ(x + θz)(1 − θ)dθπ1(dz)

and Iδc is defined as in (5.1.2) with integration over |z| > δ instead of Rd.

Definition 5.2.1. An H0-valued càdlàg adapted process u is called a solution of (5.1.1) if

(i) ut ∈ H1 for dP × dt-almost-every (ω, t) ∈ Ω × [0,T ];

(ii) E∫

]0,T ]∥ut∥

21dt < ∞;

(iii) there exists a set Ω ⊂ Ω of probability one such that for all (ω, t) ∈ [0,T ] × Ω andϕ ∈ C∞c (Rd),

(ut, ϕ)0 = (φ, ϕ)0 +

∫]0,t]

d∑i, j=1

(∂ jus, ∂−i(ai j

s ϕ))

0+ [ϕ, fs]0

ds

+

∫]0,t]

∫|z|≤δ

∫ 1

0

d∑i, j=1

(z j∂ jus(· + θz), zi∂−iϕ

)0

(1 − θ)dθπ1(dz)ds

+

∫]0,t]

∫|z|>δ

us(· + z) − us − 1[−1,1](|z|)d∑

j=1

z j∂ jus, ϕ

0

π1(dz)ds


+

∞∑ϱ=1

∫]0,t]

d∑i=0

(σiϱ

s ∂ius + gϱs , ϕ)

0dwϱ

s +

∫]0,t]

∫Rd

(us−(· + z) − us− + ot(z), ϕ)0 q(dz, ds).

Remark 5.2.2. In the above definition, instead of δ we may choose any other positiveconstant.

The following existence theorem is a consequence of Theorems 2.9, 2.10, and 4.1 in[Gyö82] and will be verified in Section 4.

Theorem 5.2.3. If Assumptions 5.2.1(m) and 5.2.2(m) hold with m ≥ 0, then there ex-ist a unique solution u of (5.1.1). Furthermore, u is a cádlág Hm-valued process withprobability one and there is a constant N = N(d,m, κ,K,T ) such that

E[supt≤T∥ut∥

2m

]+ E

∫]0,T ]∥us∥

2m+1ds ≤ Nκ2

m. (5.2.4)

Remark 5.2.4. We have used the standard definition of solution for the variational (or L2)theory fo stochastic evolution equations. In what follows below, we will always assumem ≥ 2, and so we have enough regularity to formulate the solution in the weak sense in(H1,H0,H−1) without integrating by parts.

The following proposition is needed to establish the rate of convergence in time of ourapproximation scheme and is proved in Section 4.

Proposition 5.2.5. Let Assumptions 5.2.1(m) and 5.2.2(m) hold for some m ≥ 1 and u bethe solution of (5.1.1). Moreover, assume that

supt≤T

E[∥gt∥

2m−1,ℓ2

]+ sup

t≤TE

∫Rd∥ot(z)∥2m−1π

2(dz) ≤ K.

Then there is a constant λ = λ(d,m,K,T, κ, κ2m) such that for all s, t ∈ [0,T ],

E[∥ut − us∥

2m−1

]≤ λ|t − s|.

Assumption 5.2.3 (m). For m ≥ 3, in addition to Assumption 5.2.2(m), there exists arandom variable ξ with Eξ < K such that for all ω ∈ Ω, t, s ∈ [0,T ],

∥gt∥2m−1,ℓ2

+


2(dz) ≤ ξ

∥ ft − fs∥2m−2 + ∥gt − gs∥

2m−2,ℓ2

+

∫Rd∥ot(z) − os(z)∥2m−1π

2(dz) ≤ ξ|t − s|.


Assumption 5.2.4 (m). For m ≥ 3, in addition to Assumption 5.2.1 (i), there is a constantC such that for all (ω, x) ∈ Ω × Rd, s, t ∈ [0,T ], i, j ∈ 0, 1, . . . , d,

|∂γ(ai j

t − ai js

)|2 + |∂γ

(σi

t − σis

)|2ℓ2≤ C|t − s|, ∀|γ| ≤ m − 2.

We turn our attention to the discretisation of equation (5.1.1). For each h ∈ R−0 andstandard basis vector ei, i ∈ 1, . . . , d, of Rd we define the first-order difference operatorδh,i by

δh,iϕ(x) :=ϕ(x + hei) − ϕ(x)

h,

for all real-valued functions ϕ on Rd. We define δh,0 to be the identity operator. Notice thatfor all ψ, ϕ ∈ H0, we have

(ϕ, δ−h,iψ)0 = −(δh,iϕ, ψ)0. (5.2.5)

Setδh

i :=12

(δh,i + δ−h,i)

and observe that for all ϕ ∈ H0,(ϕ, δh

i ϕ)0 = 0. (5.2.6)

For each h , 0, we introduce the grid Gh := hzk : zk ∈ Zd, k ∈ N0, z0 = 0 with step size|h|. Let ℓ2(Gh) be the Hilbert space of real-valued functions ϕ on Gh such that

∥ϕ∥2ℓ2(Gh) := |h|d∑x∈Gh

|ϕ(x)|2 < ∞.

We approximate the operators L and Nϱ by

Lht ϕ(x) :=

d∑i, j=0

ai jt (x)δh,iδ−h, jϕ(x) and Nϱ;h

t ϕ(x) :=d∑

i=0

σiϱt (x)δh,iϕ(x),

respectively. In order to approximate I, we approximate Iδ and Iδc separately. For eachk ∈ N ∪ 0 and h , 0, define the rectangles in Rd

Ahk :=

(z1

k |h| −|h|2, z1

k |h| +|h|2

]× · · · ×

(zd

k |h| −|h|2, zd

k |h| +|h|2

],

where zik, i ∈ 1, ..., d, are the coordinates of zk ∈ Zd, and set

Bhk := Ah

k ∩ |z| ≤ δ, Bhk := Ah

k ∩ |z| > δ.


We approximate Iδc by

Ihδcϕ(x) :=

∞∑k=0

(ϕ(x + hzk) − ϕ(x)) ζh,k −

d∑i=1

ξih,kδ

hi ϕ(x)

,where

ζh,k := π1(Bhk) and ξi

h,k :=∫

Bhk∩[−1,1]

ziπ1(dz).

We continue with the approximation of the operator Iδ. By (5.2.1), for all x ∈ Gh,

Iδϕ(x) =∞∑

k=0

∫Bh

k

∫ 1

0

d∑i, j=1

ziz j∂i jϕ(x + θz)(1 − θ)dθπ1(dz),

where there are only a finite number of non-zero terms in the infinite sum over k. Theclosest point in Gh to any point z ∈ Bh

k is clearly hzk. This simple observation leads us tothe following (intermediate) approximation of Iδϕ(x):

∞∑k=0

∫ 1

0

d∑i, j=1

∫Bh

k

ziz jπ1(dz)∂i jϕ(x + θhzk)(1 − θ)dθ.

However, in order to ensure that our approximation is well-defined for functions ϕ ∈ℓ2(Gh), we need to approximate the integral over θ ∈ [0, 1]. Fix k ∈ N0 and h , 0.Consider the directed line segment θhzk : θ ∈ [0, 1] extending from the origin to thepoint hzk ∈ Rd. It is clear that this line segment intersects a unique finite sequence of rect-angles from the set Ah

kk∈N0 . Denote the number of rectangles by χ(h, k). Since the line’s

start point is the origin, the first rectangle it intersects is Ah0, and since the line’s endpoint is

hzk, the last rectangle it intersects is Ahk , the center of which is the point hzk. If χ(h, k) > 2,

then in between these two rectangles, the line segment intersects χ(h, k) − 2 additionalrectangles from the set Ah

kk∈N0 − A

h0 ∪ Ah

k. Denote the indices of these rectangles by rh,kl ,

l ∈ 2, . . . , χ(h, k)−1, and set rh,k1 = 0 and rh,k

χ(h,k) = k; that is, θhzk; θ ∈ [0, 1] ⊆ ∪χ(h,k)l=1 Ah

rh,kl

.

Corresponding to the set of rectangles Ahrh,k

l

χ(h,k)l=1 is a partition 0 = θh,k

0 ≤ · · · ≤ θh,kχ(h,k) = 1

of the interval [0, 1] such that for each l ∈ 1, . . . , χ(h, k) and θ ∈ (θh,kl−1, θ

h,kl ), θhzk ∈ Ah

rh,kl

.

Since the diagonal of a d-dimensional hypercube with side length |h| has length√

dh, foreach k ∈ N0, z ∈ Bh

k , and l ∈ 1, . . . , χ(h, k),

|θz − hzrh,kl| ≤ |θz − θhzk| + |θhzk − hzrh,k

l| ≤√

d|h|, (5.2.7)


for all θ ∈ (θh,kl−1, θ

h,kl ). Set

ζi jh,k =

∫Bh

k

ziz jπ1(dz), θh,kl =

∫ θh,kl

θh,kl−1

(1 − θ)dθ

and define the operator

Ihδϕ(x) =:

∞∑k=0

χ(h,k)∑l=1

θh,kl

d∑i, j=1

ζi jh,kδh,iδ−h, jϕ(x + hzrh,k

l),

where there are only a finite number of non-zero terms in the infinite sum over k. SetIh = Ih

δ + Ihδc and introduce the martingales

ph,k,it =

∫]0,t]

∫Bh

k

ziq(dt, dz), ph,kt = q(Bh

k , ]0, t]).

Moreover, setθh,k

l := θh,kl+1 − θ

h,kl .

Let T ≥ 1 be an integer and set τ = T/T and tn = nτ for i ∈ 0, 1, . . . ,T . Forany F-martingale (pt)t≤T , we use the notation ∆pn+1 := ptn+1 − ptn . Define recursively theℓ2(Gh)-valued random variables (uh,τ

n )Tn=0 by

uh,τn (x) =uh,τ

n−1(x) +((Lh

tn−1+ Ih)uh,τ

n−1(x) + ftn−1(x))τ +

∞∑ϱ=1

(Nϱ;htn−1

uh,τn−1(x) + gϱtn−1

(x))∆wϱn

+

∞∑k=0

d∑i=1

χ(h,k)∑l=1

θh,kl δh,iuh,τ

n−1(x + hzrh,kl

)

∆ph,k,in +

∫Rd

otn−1(x, z)q(]tn−1, tn], dz)

+

∞∑k=0

(uh,τ

n−1(x + hzk) − uh,τn−1(x)

)∆ph,k

n , n ∈ 1, . . . ,T , (5.2.8)

with initial conditionuh,τ

0 (x) = φ(x), x ∈ Gh

It is clear that uh,τn is Ftn-measurable for every n ∈ 0, 1, . . . ,T . Define the operators

Lht ϕ =

d∑i, j=0

ai jt δh,iδ−h, jϕ − π

1(|z| > δ)ϕ −d∑

i=1

∫δ<|z|≤1

ziπ1(dz)δhi ϕ

and

Ihδcϕ =

∞∑k=0

ϕ(x + hzk)ζh,k

and note that Lh+ Ihδc+Iδ = Lh+Ih. On Gh, we also consider the following implicit-explicit


discretization scheme of (5.1.1):

vh,τn (x) =vh,τ

n−1(x) +((Lh

tn + Ihδ )vh,τ

n (x) + Ihδcvh,τ

n−1(x) + ftn(x))τ

+ 1n>1

∞∑ϱ=1

(Nϱ;htn−1

vh,τn−1(x) + gϱtn−1

(x))∆wϱn

+ 1n>1

∞∑k=0

d∑i=1

χ(h,k)∑l=1

θh,kl δh,ivh,τ

n−1(x + hzrh,kl

)

∆ph,k,in +

∫Rd

otn−1(x, z)q(]tn−1, tn], dz)

+ 1n>1

∞∑k=0

(vh,τ

n−1(x + hzk) − vh,τn−1(x)

)∆ ph,k

n , n ∈ 1, . . . ,T , (5.2.9)

with initial conditionvh,τ

0 (x) = φ(x), x ∈ Gh,

where 1n>1 = 0 if n = 1 and 1n>1 = 1 if n ≥ 2. A solution (vh,τn )M

n=0 of (5.2.9) is understoodas a sequence of ℓ2(Gh)-valued random variables such that vh,τ

n is Ftn-measurable for everyn ∈ 0, 1, . . . ,M and satisfies (5.1.1).

Remark 5.2.6. Under Assumptions 5.2.2 and 5.2.3, for m > 2 + d/2, by virtue of theembedding Hm−2 → ℓ2(Gh), the free-terms f , g, and o(z) are continuous ℓ2(Gh) valuedprocesses, and consequently the above schemes make sense. Moreover, for 0 < |h| < 1,there is a constant N independent of h such that -

∥ϕ∥ℓ2(Gh) ≤ N∥ϕ∥m−2. (5.2.10)

Assumption 5.2.5. The parameters h , 0 and T are such that

dτ

h2 <κ − ς(δ)(

2(supt,x,ω

∑di, j=1 |a

i jt (x)|2

)1/2+ ς1(δ)

)2 . (5.2.11)

Remark 5.2.7. We have assumed that the coefficients ai jt were bounded uniformly in ω, t,

and x, so that quantity in in denominator (5.2.11) is well-defined.

The following are our main theorems.

Theorem 5.2.8. Let Assumptions 5.2.1(m) through 5.2.4(m) hold for some m > 2 + d2 and

let Assumption 5.2.5 hold. Let u be the solution of (5.1.1) and let (uh,τn )Tn=0 be defined by

(5.2.8). Then there is a constant N = N(d,m, κ,K,T,C, λ, κ2m, δ) such that for any real

number h with 0 < |h| < 1,

E[

max0≤n≤T

supx∈Gh

|utn(x) − uh,τn (x)|2

]+ E

[max

0≤n≤T∥utn − uh,τ

n ∥2ℓ2(Gh)

]≤ N

(|h|2 + τ

).

5.3. Simulation 138

Theorem 5.2.9. Let Assumptions 5.2.1(m) through 5.2.4(m) hold for some m > 2 + d2

and let u be a solution of (5.1.1). There exists a constant R = R(d,m, κ,K, δ) such thatif T > R, then there exists a unique solution (vh,τ

n )Tn=0 of (5.2.9) and a constant N =N(d,m, κ,K,T,C, λ, κ2

m, δ) such that for any real number h with 0 < |h| < 1,

E[

max0≤n≤T

supx∈Gh

|utn(x) − vh,τn (x)|2

]+ E

[max

0≤n≤T∥utn − vh,τ

n ∥2ℓ2(Gh)

]≤ N

(|h|2 + τ

).

Remark 5.2.10. In fact, with more regularity, as consequence of Corollaries 5.5.3 and5.5.4, we can obtain stronger error estimates with difference operators in the error norms.This is an immediate consequence of the approach we use to obtaining these estimates.

5.3 Simulation

Let us consider finite difference approximations for the following SIDE on [0,T ] × Rd:

dut(x) =((σ2

1

2+σ2

2

2

)∂2

1ut(x) +∫

R(ut(x + z) − ut(x) − ∂1ut(x)z)π(dz)

)dt + σ2∂1ut(x)dwt

+

∫R

(u(x + z) − u(x)) q(dt, dz),

u0(x) =1

√2πσ0

exp(−

x2

σ21σ

20

), (5.3.1)

where π(dz) = c− exp (−β−z) dz|z|1+α− 1(−∞,0)(z) + c+ exp (−β+z) dz

|z|1+α+ 1(0,∞)(z). It is easily veri-fied that for (t, x) ∈ [0,T ] × Rd,

vt(x) =1√

π(2σ20 + 4t)

exp(

x2

σ21(σ2

0 + 2t)

)

solves

dvt(x) =σ2

1

2∂2

1vt(x)dt, v0(x) =1

√2πσ0

exp(−

x2

σ21σ

20

).

Moreover, applying Itô’s formula, we find that

ut(x) = vt

(x + σ2wt +

∫Rd

zq(dt, dz))

(5.3.2)

solves (5.3.1). Thus, we can compare our finite difference approximations with (5.3.2).In our numerical simulations, we used MATLAB 2013a and made the following pa-

5.3. Simulation 139

rameter specification:

σ1 =12, σ2 =

14, σ0 =

12, c− = c+ = 1, β− = β+ = 1, α− = α+ = 1.1, T = 1.

We also made a few practical simplifications. Both the explicit and implicit-explicit ap-proximations were assumed to take the value zero on (−∞, 8] ∪ [8,∞). We also restrictedthe support of π(dz) to [−3, 3]. We would like to investigate the associated error with thesereductions in the future. Regarding the first reduction, we mention that a good heuristic isto choose the size of domain according to the support of the free terms and the exit timeof the diffusion for which the drift of the SIDE (up to zero order terms) is the infinitesimalgenerator of. In fact, it is more than a heuristic and we aim to address this in a future work.

In our simulation, we took δ = 1100 . It follows that κ = σ2

1 =12 and

ς(δ) = c−

∫ δ

0exp(−β−z)z1−α−dz + c+

∫ δ

0exp(−β+z)z1−α+dz + z

= c−βα−−2− γ(2 − α−, β−δ) + c+β

α+−2+ γ(2 − α+, β+δ) ≈ 0.0082,

where γ(η, z) denotes the lower incomplete gamma function. Thus, the right-hand-sideof (5.2.11) is approximately 1.0559, and hence we can always set τ = h2. The quan-tities ζ11

h,k, ζh,k, and ξ1h,k can all be calculated using MATLAB’s built-in upper and lower

incomplete gamma functions, or by implementing an appropriate numerical integrationprocedure. The calculation of θh,k

l , θh,kl , and θh,k

l are all straightforward in one-dimension.Some more thought would need to spent on how to calculate these quantities in higherdimensions. Of course as an alternative, one could set δ = h

2 , but then the schemes arenot guaranteed to converge as h tends to zero. This is the drawback of taking δ = h

2 andnot including the additional terms in Iδ (see the paragraph at the bottom of page 1620 in[CV05]). It does seem that the method we propose to discretise Iδ is novel in this respect.In our error analysis, we have considered h ∈ 2−2, 2−3, 2−4, 2−5, 2−6, 2−7 and τ = h2.

The term ∫|z|>δ

(ut(x + z) − ut(x)) π(dz)

in the drift of (5.3.1) can be cancelled with the compensator of the compensated Poissonrandom measure term. We get a similar cancellation in the corresponding finite differenceequations, and thus we can replace ph,k

t = q(Bhk , ]tn, tn+1]) with ph,k

t = p(Bhk , ]tn, tn+1]) in the

explicit 5.2.8 and implicit-explicit (5.2.9) scheme.In order to simulate

∆ph,kn =

∫]tn,tn+1]

∫Bh

k

zq(dt, dz), ph,kt = p(Bh

k , ]tn, tn+1]),

5.3. Simulation 140

for the finest time step size τ = 2−14, we used the algorithm discussed in Section 4of [KM11]. In this algorithm, a parameter ϵ is chosen for which the process ∆ph,0

n =∫]0,t]

∫|z|<ϵ

zq(dt, dz) is approximated by a Wiener process with infinitesimal variance∫|z|<ϵ

z2π(dz). We chose the parameter ϵ = 2−8, which is one-half times the smallest step size hunder consideration in our error analysis. The process

∫]0,t]

∫|z|>ϵ

zq(dt, dz) =∫

]0,t]

∫|z|>ϵ

zp(dt, dz)(we have used symmetry of the measure π(dz))) is a compound Poisson process with jumpintensity

λ := 2∫ 3

ϵ

π(dz) ≈ 68.9676

and jump-size density

f (z) =1λ

(c− exp (−β−z)

dz|z|1+α−

1(−3,2−8)(z) + c+ exp (−β+z)dz|z|1+α+

1(2−8,3)(z)).

The underlying Poisson process was simulated using MATLAB’s built-in Poisson randomvariable generator; of course there are other simple methods that one can use as an alter-native (e.g. exponential times or uniform times for fixed number of jumps). We sampledrandom variables from the density f by sampling the positive and negative parts separatelyand using an acceptance-rejection algorithm with a Pareto random variable. We refer to[KM11] for more details. Once we simulated the point process on [0,T ]× [−3,−ϵ]∪[ϵ, 3],we then computed

∫]0,t]

∫|z|>ϵ

zp(dt, dz). In order to compute ph,kt = p(Bh

k , ]tn, tn+1]), we rana histogram with the intervals Bh

k .The quantity ∆ph,k

n =∫

]tn,tn+1]

∫Bh

kzq(dt, dz) is zero for k , 0 when h < δ

2 (for h ∈

2−2, 2−3, 2−4, 2−5) since Bhk = ∅ for k , 0 when h < δ

2 . For h ∈ 2−6, 2−7, ∆ph,kn is

non-zero for k ∈ −1, 0, 1. A similar analysis holds for the quantity ζ11h,k. As mentioned

above, we set ph,0t equal to the Weiner process approximating the small jumps. To compute∫

]tn,tn+1]

∫Bh

kzq(dt, dz) for k ∈ −1, 1 in the case h ∈ 2−6, 2−7, we summed the jump sizes in

their respective bins and compensated. To obtain the above quantities for coarser time stepsizes, we cumulatively summed the finer increments and took the union of jump sizes.

Lastly, we made use of the Fast Fourier Transform to compute terms of the form

∞∑k=0

ϕ(x + hzk)∆ ph,kn ,

which would be quite computationally expensive otherwise. In our error analysis, we ran3000 simulations of the explicit and implicit-explicit schemes on 30 CPUs and computedthe following quantities:

By our main theorems and the relation τ = h2, these errors should proportional to h(i.e. O(h)). This is precisely what we observe in Figure 5.1. The slight bump down at thefinest two spatial step-sizes h ∈ 2−6, 2−7 is most likely due to the increase in the number

5.4. Auxiliary results 141

-7 -6 -5 -4 -3 -2−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

log2(h = τ2)

log 2

(err

or(h

))

Rate of convergence (3000 simulations)

√∑3000m=1 max0≤n≤T supx∈Gh

|utn(x) − uh,τn (x)|2√∑3000

m=1 max0≤n≤T ∥utn − uh,τn ∥

2ℓ2(Gh)√∑3000

m=1 max0≤n≤T supx∈Gh|utn(x) − vh,τ

n (x)|2√∑3000m=1 max0≤n≤T ∥utn − vh,τ

n ∥2ℓ2(Gh)

Line of slope 1

Figure 5.1 Simulated errors with respect to the space discretization and a line as referenceslope on a log2 scale

.

of terms in the approximation of Ihδ (three to be precise) and the analogous small jump

term in the noise.

5.4 Auxiliary results

In this section, we present some results that will be needed for the proof of Theorems 5.2.8and 5.2.9. Introduce the operators

Iδ;h(z)ϕ(x) :=∞∑

k=0

1Bhk(z)

χ(h,k)∑l=1

d∑i=1

θh,kl ziδh,iϕ(x + hzrh,k

l),

Iδc;h(z)ϕ(x) :=

∞∑k=0

1Bhk(z)(ϕ(x + hzk) − ϕ(x)),

Ih(z)ϕ(x) := Iδ;h(z)ϕ(x) + Iδc;h(z)ϕ(x).

Consider the following explicit and implicit-explicit schemes in H0:

uh,τn =uh,τ

n−1 +((Lh

tn−1+ Ih)uh,τ

n−1 + ftn−1

)τ +

∞∑ϱ=1

(Nϱ;htn−1

uh,τn−1 + gϱtn−1

)∆wϱn

+

∫Rd

(Ih(z)uh,τ

n−1 + otn−1(z))

q(dz, ]tn−1, tn]), n ∈ 1, . . . ,T , (5.4.1)


and

vh,τn =vh,τ

n−1 +((Lh

tn + Ihδ )vh,τ

n + Ihδcvh,τ

n−1 + ftn

)τ + 1n>1

∞∑ϱ=1

(Nϱ;htn−1

vh,τn−1 + gϱtn−1

)∆wϱn

+ 1n>1

∫Rd

(Ih(z)vh,τ

n−1 + otn−1(z))

q(dz, ]tn−1, tn]), n ∈ 1, . . . ,T , (5.4.2)

with initial conditionuh,τ

0 (x) = vh,τ0 (x) = φ(x), x ∈ Rd.

We now prove some lemmas that will help us to establish the consistency of our approxi-mations. The following lemma is well-known and we omit the proof (see, e.g., [GK10]).

Lemma 5.4.1. For each integer m ≥ 0, there is a constant N = N(d,m) such that for allu ∈ Hm+2 and v ∈ Hm+3,

∥δh,iu − ∂iu∥m ≤12|h|∥u∥m+2,

∥δh,iδ−h, jv − ∂i jv∥m ≤ N|h|∥v∥m+3.

Lemma 5.4.2. For each integer m ≥ 0, there is a constantN = N(d,m, δ) such that for all u ∈ Hm+3, we have

∥Iu − Ihu∥m ≤ N|h|∥u∥m+3. (5.4.3)

Proof. It suffices to show (5.4.3) for u ∈ C∞c (Rd). We begin with m = 0. A simplecalculation shows that

Iδcu(x) − Ihδcu(x) =

∞∑k=0

∫Bh

k

∫ 1

0

d∑i=1

(zi − hzik)∂iu(x + hzk + θ(z − hzk))dθπ1(dz)

−

∞∑k=0

∫Bh

k∩[−1,1]

d∑i=1

zi(∂iu(x) − δhi u(x))π1(dz).

By Minkowski’s inequality, we get

∥Iδcu − Ihδcu∥0 ≤

∞∑k=0

∫Bh

k

d∑i=1

|zi − hzik|∥∂iu∥0π1(dz)

+

∞∑k=0

∫Bh

k∩[−1,1]

d∑i=1

|zi|∥∂iu(x) − δhi u(x)∥0π1(dz) ≤ N|h|∥u∥3 + N

d∑i=1

∥∂iu(x) − δhi u(x)∥0,

since |z − hzk| ≤ |h|√

d/2 and (5.2.3) holds. Thus, by Lemma 5.4.1, we have

∥Iδcu − Ihδcu∥0 ≤ N |h|∥u∥3. (5.4.4)


We also haveIδu(x) − Ih

δu(x) =

∞∑k=0

∫Bh

k

χ(h,k)∑l=1

∫ θh,kl

θh,kl−1

d∑i, j=1

ziz j(∂i ju(x + θz) − δh,iδ−h, ju(x + hzrh,k

l))

(1 − θ)dθπ1(dz). (5.4.5)

Note that

∂i ju(x + θz) − δh,iδ−h, ju(x + hzrh,kl

)

= ∂i ju(x + θz) − ∂i ju(x + hzrh,kl

) + ∂i ju(x + hzrh,kl

) − δh,iδ−h, ju(x + hzrh,kl

)

=

∫ 1

0

d∑q=1

(θzq − hzq

rh,kl

)∂q∂i ju

(x + hzrh,k

l+ ρ(θz − hzrh,k

l))

dρ

+∂i ju(x + hzrh,kl

) − δh,iδ−h, ju(x + hzrh,kl

).

By (5.2.7), we have |θzq − hzq

rh,kl

| ≤ N|h|. Hence, substituting the above relation in (5.4.5),using Minkowski’s inequality, (5.2.2), and Lemma 5.4.1, we obtain

∥Iδu − Ihδu∥0 ≤ |h|N∥u∥3. (5.4.6)

Combining (5.4.4) and (5.4.6), we have (5.4.3) for m = 0. The case m > 0 follows fromthe case m = 0, since for a multi-index γ, we have

∂γ(Iu − Ihu) = I∂γu − Ih∂γu.

Lemma 5.4.3. For each integer m ≥ 0, there is a constant N = N(d,m, δ), there is aconstant such that for all u ∈ Hm+2, we have∫

Rd∥Ih(z)u − I(z)u∥2mπ

2(dz) ≤ N|h|2∥u∥2m+2. (5.4.7)

Proof. It suffices to prove the lemma for u ∈ C∞c (Rd) and m = 0. We have

Iδ(z)u(x) − Iδ;h(z)u(x) =∞∑

k=0

1Bhk(z)

χ(h,k)∑l=1

∫ θh,kl

θh,kl−1

d∑i=1

zi(∂iu(x + θz) − δh,iu(x + hzrh,kl

))dθ.


Notice that

∂iu(x + θz) − δh,iu(x + hzrh,kl

) =∫ 1

0

d∑i, j=1

∂i ju(x + ρ(θz − hzrh,kl

))(θz j − hz j

rh,kl

)dρ

+∂iu(x + hzrh,kl

) − δh,iu(x + hzrh,kl

).

Thus, by Remark 5.2.7 and Lemma 5.4.1, we get

∥Iδ;h(z)u − Iδ(z)u∥20 ≤ 1|z|≤δ|z|2N|h|2∥u∥22,

and hence by (5.2.2), we obtain∫Rd∥Iδ;h(z)u − Iδ(z)u∥20π

2(dz) ≤ N |h|2∥u∥22. (5.4.8)

We also have

|Iδc(z)u(x) − Iδ

c;h(z)u(x)| =∞∑

k=0

1Bhk(z)|u(x + z) − u(x + hzk)|

≤

∞∑k=0

1Bhk(z)

∫ 1

0

d∑i=1

|∂iu(x + hzk + ρ(z − hzk))∥zi − hzik|dρ.

Consequently,∥Iδ

c;h(z)u − Iδc(z)u∥20 ≤ 1|z|>δN |h|2∥u∥21,

which implies by (5.2.3) that∫Rd∥Iδ

c;h(z)u − Iδc(z)u∥20π

2(dz) ≤ N |h|2∥u∥21. (5.4.9)

Combining (5.4.9) and (5.4.8), we have (5.4.7) for m = 0. The case m > 0 follows fromthe case m = 0, since for a multi-index γ, we have

∂γ(Iu − Ihu) = I∂γu − Ih∂γu.

Lemma 5.4.4. If Assumption 5.2.1(m) holds for some m ≥ 0, then for any ϵ ∈ (0, 1) thereexists constants N1 = N1(d,m, κ,K, δ, ϵ) and N2 = N2(d,m, κ,K, δ, ϵ) such that for anyu ∈ Hm,

G(m)t (u) := 2(u,Lh

t u)m + ∥Nht u∥2m,ℓ2

+ 2(u, Ihu)m +

∫Rd∥Ih(z)u∥2mπ

2(dz)


≤ −(κ − ς(δ) − ϵ)d∑

i=1

∥δh,iu∥2m + N1∥u∥2m, (5.4.10)

and

(u, Lht u)m + (u, Ih

δu)m ≤ −(κ − ς1(δ) − ϵ)d∑

i=1

∥δh,iu∥2m + N2∥u∥mn . (5.4.11)

Proof. By virtue of Lemma 3.1 and Theorem 3.2 in [GK10], under Assumption 5.2.1,there is a constant N = N(d,m, κ) such that for any u ∈ Hm and ϵ > 0,

2(u,Lht u)m + ∥N

ht u∥2m,ℓ2

≤ −(κ − ϵ)d∑

i=1

∥δh,iu∥2m + N∥u∥2m.

Therefore, it suffices to show that there is a constant N = N(δ) such that for all u ∈ C∞c (Rd),

2(u, Ihu)m +


2(dz) ≤ ς(δ)d∑

i=1

∥δh,iu∥2m + N∥u∥2m. (5.4.12)

We start with m = 0. Since

(u, Ihδu)0 =

∞∑k=0

∫Bh

k

χ(h,k)∑l=1

d∑i, j=1

θk,hl ziz j

∫Rdδh,iδ−h, ju(x + hzrh,k

l)u(x)dxπ1(dz)

and ∫Rdδh,iδ−h, ju(x + hzrh,k

l)u(x)dx = −

∫Rdδh,iu(x + hzrh,k

l)δh, ju(x)dx,

by Hölder’s inequality, we get

2(u, Ihδu)0 ≤

∫|z|≤δ|z|2π1(dz)

d∑i=1

∥δh,iu∥20 = ς1(δ)d∑

i=1

∥δh,iu∥20.

In addition, owing to Holder’s inequality and (5.2.6), we have

2(u, Ihδcu)0 =

∞∑k=0

∫Bh

k

∫Rd

u(x + hzk) − u(x) − 1[−1,1](z)d∑

i=1

ziδhi u(x)

u(x)dxπ1(dz) ≤ 0.

By Minkowski’s inequality, we have

∥Iδ;h(z)u∥2 ≤∞∑

k=0

1Bhk(z)|z|2

d∑i=1

∥δh,iu∥20 and ∥Iδc;h(z)u∥20 ≤ 4

∞∑k=0

1Bhk(z)∥u∥20

and hence ∫Rd∥Ih(z)u∥20π

2(dz) ≤ ς2(δ)d∑

i=1

∥δh,iu∥20 + 4π1(|z| > δ)∥u∥20,


which proves (5.4.12) for m = 0. The case m > 0 follows by replacing u with ∂γu for|γ| ≤ m. This proves (5.4.10), which implies (5.4.11).

Remark 5.4.5. It follows that for m ≥ 0, there is a constant N5 = N5(d,m,K, δ) such thatfor any u ∈ Hm,

∥Nϱ;ht u∥2m +


2(dz) ≤ N5

d∑i=0

∥δh,iu∥2m (5.4.13)

≤ N5

(1 +

4dh2

)∥u∥2m. (5.4.14)

Lemma 5.4.6. For any m ≥ 0 and u ∈ Hm,

∥Ihδcu∥2m ≤ π

1(|z| > δ)2∥u∥2m. (5.4.15)

Moreover, if Assumption 5.2.1 holds for some m ≥ 0, then for any ϵ > 0 and u ∈ Hm,

∥(Lht + Ih)u∥2m ≤ (1 + ε)

N3dh2

d∑i=1

∥δh,iu∥2m + N4

(1 +

1h2

)∥u∥2m (5.4.16)

where

N3 :=

2sup

t,x,ω

d∑i, j=1

|ai j(x)|2

1/2

+ ς1(δ)

2

and N4 is a constant depending only on d,m,K, δ, and ϵ.

Proof. It suffices to prove the lemma for u ∈ C∞c (Rd). It follows that

(Lht + Ih

δ )u(x) =∞∑

k=0

χ(h,k)∑l=1

θh,kl

d∑i, j=1

ζi jt,h,k(x)δh,iδ−h, ju(x + hzrh,k

l) +

d∑i, j=0

i or j=0

ai jt δh,iδ−h, ju(x)

where ζ i jt,h,k(x) := ζ

i jh,k for k , 0 and ζ

i jt,h,0(x) := ζ

i jh,0 + 2ai j

t (x) (recall that θh,01 = 1

2 andχ(h, 0) = 1). Moreover, for each multi-index γ with 1 ≤ |γ| ≤ m,

∂γ(Lht + Ih

δ )u(x) =∞∑

k=0

χ(h,k)∑l=1

θh,kl

d∑i, j=1

ζi jh,k(x)δh,iδ−h, j∂

γu(x + hzrh,kl

)

+∑β : β<γ

N(β, γ)d∑

i, j=1

(∂γ−βai j

t (x))δh,iδ−h, j∂

βu(x)

+∑β : β≤γ

N(β, γ)d∑

i, j=0i or j=0

((∂γ−βai j

t (x))δh,iδ−h, j∂

βu(x))


=: (A1(γ) + A2(γ) + A3(γ))u(x),

where N(β, γ) are constants depending only on β and γ. By Young’s inequality andJensen’s inequality, for any ϵ ∈ (0, 1), we have

∥(Lht + Ih)u∥2m ≤ (1 + ϵ)

∑|γ|≤m

∥A1(γ)u∥20

+3(1 +

1ϵ

) ∑|γ|≤m

(∥A2(γ)u∥20 + ∥A3(γ)u∥20

)+ ∥Ih

δcu∥2m

.Applying Minkowski’s inequality and the Cauchy-Bunyakovsky-Schwarz inequality andnoting that

∑χ(h,k)l=1 θh,k

l =12 and

||δh,i∂βu||0 ≤

2h||∂βu||0; ∀i ∈ 0, 1 . . . , d, ∀|β| = m,

we obtain

∥A1(γ)u∥0 ≤∞∑

k=0

χ(h,k)∑l=1

θh,kl

supt,x,ω

d∑i, j=1

∣∣∣ζ i jh,k(x)

∣∣∣21/2 d∑

i, j=1

∣∣∣∣∣∣∣∣δh,iδ−h, ju(· + hzrh,kl

)∣∣∣∣∣∣∣∣2

m

1/2

≤

√d

h

∞∑k=0

supt,x,ω

d∑i, j=1

|ζi jh,k(x)|2

1/2 d∑

i=1

∥δh,i∂γu∥20

1/2

and

∞∑k=0

supt,x,ω

d∑i, j=1

|ζi jh,k(x)|2

1/2

=

supt,x,ω

d∑i, j=1

∣∣∣∣∣∣∫

Bh0

ziz jπ1(dz) + 2ai jt (x)

∣∣∣∣∣∣2

1/2

+

∞∑k=1

d∑i, j=1

∣∣∣∣∣∣∫

Bhk

ziz jπ1(dz)

∣∣∣∣∣∣2

1/2

≤ 2

supt,x,ω

d∑i, j=1

|ai jt (x)|2

1/2

+ ς(δ).

Thus, ∑|γ|≤m

||A1(γ)u||20 ≤N3dh2

d∑i=1

∥∂h,iu∥2m.

Another application of the Cauchy-Bunyakovsky-Schwarz inequality andMinkowski’s inequality, combined with the inequalities

||δh,i∂βu||0 ≤ ||∂i∂

βu||0 ∀i ∈ 0, 1 . . . , d, ∀|β| ≤ m − 1,

||δh,iδ−h, j∂βu||0 ≤ ||∂i j∂

βu||0, ∀; i, j ∈ 1, . . . , d, ∀|β| ≤ m − 2,


and||δh,iδ−h, j∂

βu||0 ≤2h∥δh,iu∥m, ∀; i, j ∈ 1, . . . , d, ∀|β| = m − 1,

yields ∑|γ|≤m

(∥A2(γ)u∥20 + ∥A3(γ)∥20

)≤ N

(1 +

1h2

)||u||2m .

By Minkowski’s integral inequality, we have

∥Ihδcu∥m ≤

∫Rd

∞∑k=0

1B

hk∥u(· + hzk) − u − 1[−1,1](z)

d∑i=1

ziδh,iu∥mπ1(dz)

≤ 3

π1(|z| > δ) +2d

∫δ<|z|≤1

|z|π1(dz)

h

∥u∥m.It is also easy to see that (5.4.15) holds. Combining above inequalities, we obtain (5.4.16).

The following theorem establishes the stability of the explicit approximate scheme(5.4.1).

Theorem 5.4.7. Let Assumption 5.2.1 hold with m ≥ 0 and Assumption 5.2.5 hold. LetF i ∈ Hm for i ∈ 0, ..., d, G ∈ Hm(ℓ2), and R ∈ Hm(π2). Consider the following scheme inHm:

uh,τn = uh,τ

n−1 +

∫]tn−1,tn]

(Lhtn−1+ Ih)uh,τ

n−1 +

d∑i=0

δh,iF it

dt +∫

]tn−1,tn]

(N

ϱ;htn−1

uh,τn−1 +Gϱ

t

)dwϱ

t

+

∫]tn−1,tn]

∫Rd

(Ih(z)uh,τ

n−1 + Rt(z))

q(dt, dz), n ∈ 1, . . . ,T ,

for any Hm−valued F0−measurable initial condition φ. If ϕ ∈ Hm(F0), then there is aconstant N = N(d,m, κ,K,T, δ) such that

E[

max0≤n≤T

∥uh,τn ∥

2m

]+ E

T∑n=0

τ

d∑i=0

∥δh,iuh,τn ∥

2m ≤ NE

[∥φ∥2m

]

+NE∫ T

0

( d∑i=0

∥F it∥

2m + ∥Gt∥

2m +

∫Rd∥Rt(z)∥2mπ

2(dz))dt. (5.4.17)

Proof. If E[∥φ∥2m

]< ∞, then proceeding by induction on n and using Young’s and Jensen’s

inequality, Itô’s isometry, (5.4.16), and (5.4.14), we get that for all n ∈ 0, 1, . . . ,T ,E

[∥uh,τ

n ∥2m

]< ∞. Applying the identity ∥y∥2m − ∥x∥

2m = 2(x, y − x)m + ∥y − x∥2m, x, y ∈ Hm,


for each n ∈ 1, . . . ,T , we obtain

∥uh,τn ∥

2m = ∥u

h,τn−1∥

2m +

6∑i=1

Ii(tn), (5.4.18)

where

I1(tn) := 2τ(uh,τn−1,

(Lh

tn−1+ Ih

)uh,τ

n−1)m + ∥η(tn)∥2m,

I2(tn) := 2∫

]tn−1,tn]

d∑i=0

(uh,τn−1, δh,iF i

t)mdt,

I3(tn) :=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣τ (Lh

tn−1+ Ih

)uh,τ

n−1 +

∫]tn−1,tn]

d∑i=0

δh,iF itdt

∣∣∣∣∣∣∣∣∣∣∣∣∣∣2

m

,

I4(tn) := 2∫

]tn−1,tn]

(uh,τ

n−1,Nϱ;htn−1

uh,τn−1 +Gϱ

t

)m

dwϱt ,

I5(tn) := 2∫

]tn−1,tn]

∫Rd

(uh,τ

n−1,Ih(z)uh,τ

n−1 + Rt(z))

mq(dt, dz),

I6(tn) := 2(τ(Lh

tn−1+ Ih)uh,τ

n−1, η(tn))

m+ 2

∫]tn−1,tn]

d∑i=0

δh,iF itdt, η(tn)

m

,

and where

η(tn) :=∫

]tn−1,tn]

(N

ϱ;htn−1

uh,τn−1 +Gϱ

t

)dwϱ

t +

∫]tn−1,tn]

∫Rd

(Ih(z)uh,τ

n−1 + Rt(z))

q(dt, dz).

By virtue of Assumption 5.2.5, we fix q > 0 and ϵ > 0 small enough such that

q := κ − ς(δ) − ϵ − (1 + ϵ)(1 + q)N3dτ

h2 − q > 0,

where N3 is the constant in (5.4.6). Since the two stochastic integrals that define η areorthogonal square-integrable martingales, by Young’s inequality and (5.4.13), for all q > 0,

E[∥η(tn)∥2m

]≤ E

[τ∥Nh

tn−1uh,τ

n−1∥2m,ℓ2

]+ Eτ

∫Rd∥Ih(z)uh,τ

n−1∥2mπ

2(dz) + qEτd∑

i=0

∥δh,iuh,τn−1∥

2m

+

(1 +

N5

q

)E

∫]tn−1,tn]

(∥Gt∥

2m,ℓ2+


2(dz))

dt. (5.4.19)

Thus, taking q = q3 in (5.4.19), we have

EI1(tn) ≤ EτG(m)tl−1

(uh,τl−1) +

q3

Eτd∑

i=0


2m


+

(1 +

3N5

q

)E

∫]tn−1,tn]

(∥Gt∥

2m,ℓ2+


2(dz))

dt.

Using (5.2.5) and Young’s inequality, we obtain

EI2(tn) ≤q3

Eτd∑

i=0


2m +

3q

E∫

]tn−1,tn]

d∑i=0

∥F it∥

2mdt.

An application of Young’s inequality and (5.4.16) yields

EI3(tn) ≤ (1 + ϵ)(1 + q)N3dτ

h2 Eτd∑

i=1

∥δh,iuh,,τn−1∥

2m + (1 + q)N4

(τ +

τ

h2

)E

[τ∥uh,τ

n−1∥2m

]+(d + 1)

(1 +

1q

)E

∫]tn−1,tn]

τ∥F0t ∥

2m +

4dτh2

d∑i=1

∥F it∥

2m

dt.

Making use of the estimate (5.4.14) and noting that E∥uh,τn ∥

2m < ∞, G ∈ Hm(ℓ2), and

R ∈ Hm(π2), we obtain EI4(tn) = EI5(tn) = 0. Moreover, as (Lhtn−1+ Ih)uh,τ

n−1 is Ftn−1-measurable and E(η(tn)|Ftn−1) = 0, the expectation of first term in I6(tn) is zero, and henceby Young’s inequality, for any q1 > 0,

EI6(tn) ≤ q1E[∥η(tn)∥2m

]+

1q1

E

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∫

]tn−1,tn]

d∑i=0

δh,iF itdt

∣∣∣∣∣∣∣∣∣∣∣∣∣∣2

m

.

Moreover, by Jensen’s inequality, (5.4.19), and (5.4.13), for any q1 > 0 and q > 0,

EI6(tn) ≤ (q1q + q1N5)Eτd∑

i=0


2m

+E∫

]tn−1,tn]

(d + 1)τq1

∥F0t ∥

2m +

4d(d + 1)τq1h2

d∑i=1

∥F it∥

2m

dt

+q1

(1 +

N5

q

)E

∫]tn−1,tn]

(∥Gt∥

2m,ℓ2+


2(dz))

dt.

We choose q and q1 such that q1q + q1N5 ≤ q/3. Thus, owing to (5.4.10), we have

EG(m)tn−1

(uh,τn−1) +

(q + (1 + ϵ)(1 + q)N3d

τ

h2

)Eτ

d∑i=1


2m

≤ −qEτd∑

i=1


2m + N1E

[τ||uh,τ

n−1||2m

].

Taking the expectation of both sides of (5.4.18), summing-up, and combining the above


inequalities and identities, we find that there is a constant N = N(d,m, κ,K, δ) such thatfor all n ∈ 0, 1, . . . ,T ,

E[∥uh,τ

n ∥2m

]≤ E

[∥φ∥2m

]− qE

n∑l=1

τ

d∑i=1

∥δh,iuh,τl−1∥

2m

+

(N1 + q + (1 + q)N4

(τ +

τ

h2

))E

n∑l=1

τ∥uh,τl−1∥

2m

+N(τ +

τ

h2

)E

∫]0,tn]

d∑i=0

∥F it∥

2mdt + NE

∫]0,tn]

(∥Gt∥

2m,ℓ2+


2(dz))

dt.

Therefore, by discrete Gronwall’s inequality, there is a constant N = N(d,m, κ,K,T, δ)such that

E[∥uh,τ

n ∥2m

]+ E

n∑l=0

τ

d∑i=0

∥δh,iuh,τl ∥

2m ≤ NE

[∥φ∥2m

]

+NE∫

]0,T ]

d∑i=0

∥F it∥

2m + ∥Gt∥

2m,ℓ2+


2(dz)

dt. (5.4.20)

Now that we have proved (5.4.20), we will show (5.4.17). Estimating as we did above, weget that there is a constant N such that

E max0≤n≤T

n∑l=1

(I1(tl) + I2(tl) + I3(tl) + I6(tl)) ≤ NET−1∑l=0

τ

d∑i=0

∥δh,iuh,τl ∥

2m

+NE∫

]0,T ]

d∑i=0

∥F it∥

2m + ∥Gt∥

2m,ℓ2+


2(dz)

dt.

Applying the Burkholder-Davis-Gundy inequality and Young’s inequality, we obtain

E max0≤n≤T

n∑l=1

I5(tl) ≤ 6E

∣∣∣∣∣∣∣n∑

l=1

∫]tn−1,tn]

∫Rd

(uh,τ

n−1,Ih(z)uh,τ

n−1 + Rt(z))2

mπ2(dz)dt

∣∣∣∣∣∣∣1/2

≤14

E[

max0≤n≤T

∥uh,τn ∥

2m

]+ N

E T−1∑l=0

τE∥δh,iuh,τl ∥

2m + E

T−1∑l=0

τE∥uh,τl ∥

2m

+NE

∫]0,T ]


2(dz)dt.

We can estimate E max0≤n≤T∑n

l=1 I4(tl) in similar way. Combining the above E max0≤n≤T -estimates and (5.4.20), we obtain (5.4.17).

The following theorem establishes the existence and uniqueness of a solution to (5.4.2)and the stability of the implicit-explicit approximation scheme.


Theorem 5.4.8. Let Assumption 5.2.1 hold with m ≥ 0. Let F i ∈ Hm for i ∈ 0, ..., d,G ∈ Hm(ℓ2) and R ∈ Hm(π2). Then there exists a constant R = R(d,m, κ,K, δ) such that ifT > R, then for any h , 0, there exists a unique Hm-valued solution (vh,τ

n )Tn=0 of

vh,τn = vh,τ

n−1 +

∫]tn−1,tn]

(Lhtn + Ih

δ )vh,τn + Ih

δcvh,τn−1 +

d∑i=0

δh,iF it

dt

+

∫]tn−1,tn]

(1n>1N

ϱ;htn−1

vh,τn−1 +Gϱ

t

)dwϱ

t

+

∫]tn−1,tn]

∫Rd

(1n>1I

h(z)vh,τn−1 + Rt(z)

)q(dt, dz), (5.4.21)

for n ∈ 1, . . . ,T , for any Hm−valued F0−measurable initial condition φ. Moreover, ifϕ ∈ Hm(F0), then there is a constant N = N(d,m, κ,K,T, δ) such that

E[

max0≤n≤T

∥vh,τn ∥

2m

]+ E

T∑n=0

τ

d∑i=0

∥δh,ivh,τn ∥

2m ≤ NE

[∥φ∥2m

]

+NE∫ T

0

d∑i=0

∥F it∥

2m + ∥Gt∥

2m +


2(dz)

dt. (5.4.22)

Proof. For each n ∈ 1, . . . ,T , we write (5.4.21) as

Dnvh,τn = yn−1,

where Dn is the operator defined by

Dnϕ := ϕ − τ(Lh

tn + Ihδ

)ϕ

and

yn−1 := vh,τn−1 +

∫]tn−1,tn]

Ihδcvh,τ

n−1 +

d∑i=0

δh,iF it

dt +∫

]tn−1,tn]

(1n>1N

ϱ;htn−1

vh,τn−1 +Gϱ

t

)dwϱ

t

+

∫]tn−1,tn]

∫Rd

((1n>1I


)q(dt, dz).

Fix ϵ1 and ϵ2 in (0, 1) such that

q1 := κ − ς1(δ) − ϵ1 > 0.

andq2 := κ − ς(δ) − ϵ2 > 0.


Owing to Lemma 5.4.6, there is a constant N = N(d,m,K, δ) such that for all ϕ ∈ Hm,

∥Dnϕ∥2m ≤ N

(1 + τ2

(1h2 +

1h4

))∥ϕ∥2m. (5.4.23)

Assume T > T N2. By (5.4.11), for all ϕ ∈ Hm, we have

(ϕ,Dnϕ)m ≥ (1 − τN2)∥ϕ∥2m + q1τ

d∑i=1

∥δh,iϕ∥2m ≥ (1 − τN2)∥ϕ∥2m. (5.4.24)

Using Jensen’s inequality and (5.4.15), we get

∥y0∥2m ≤ 5

(1 + π1(|z| > δ)2τ2

)∥ϕ∥2m +

20τh2

∫]0,t1]

d∑i=0

∥F it∥

2mdt + 5

∣∣∣∣∣∣∣∣∣∣∣∣∫

]0,t1]Gϱ

t dwϱt

∣∣∣∣∣∣∣∣∣∣∣∣2m

+ 5

∣∣∣∣∣∣∣∣∣∣∣∣∫

]0,t1]

∫Rd

Rt(z)q(dt, dz)

∣∣∣∣∣∣∣∣∣∣∣∣2m

. (5.4.25)

Since φ ∈ Hm, F i ∈ Hm, i ∈ 0, 1, . . . , d, G ∈ Hm(ℓ2), and R ∈ Hm(π2), it follows thaty0 ∈ Hm. By (5.4.23), and (5.4.24), owing to Proposition 3.4 in [GM05] (p = 2), thereexists a unique vh,τ

1 in Hm such that D1vh,τ1 = y0, and moreover

∥vh,τ1 ∥

2m ≤ 1 +

∥y0∥2m

(1 − τN2)2 < ∞. (5.4.26)

Proceeding by induction on n ∈ 1, . . . ,T , one can show that there exists a unique vh,τn in

Hm such that Dnvh,τn = yn−1, and moreover

∥vh,τn ∥

2m ≤ 1 +

∥yn−1∥2m

(1 − τN2)2 < ∞. (5.4.27)

Assume that E∥φ∥2m < ∞. By (5.4.25) and (5.4.26) and the fact that f i ∈ Hm, i ∈0, 1, . . . , d, g ∈ Hm(ℓ2), and R ∈ Hm(π2), it follows that E∥vh,τ

1 ∥2m < ∞. By Jensen’s

inequality, (5.4.15), and (5.4.14), we have

E[∥yn−1∥

2m

]≤ 7N

(1 + π1(|z| > δ)2τ2 + 1n>1τ

(1 +

1h2

))E

[∥vh,τ

n−1∥2m

]+

28τh2 E

∫]0,t1]

d∑i=0

∥F it∥

2mdt + 7E

∫]0,t1]∥Gt∥

2m,ℓ2

dt + 7E∫

]0,t1]


2(dz)dt.(5.4.28)

Proceeding by induction on n and combining (5.4.27) and (5.4.28), we obtain

E[∥vh,τ

n ∥2m

]< ∞, ∀n ∈ 0, 1, . . . ,T . (5.4.29)


Applying the identity ∥y∥2m−∥x∥2m = 2(x, y−x)m+∥y−x∥2m, x, y ∈ Hm, for any n ∈ 1, . . . ,T ,

we have

∥vh,τn ∥

2m = ∥v

h,τn−1∥

2m +

6∑i=1

Ii(tn),

where

I1(tn) := 2τ(vh,τn ,

(Lh

tn + Ihδ

)vh,τ

n )m + 2τ(vh,τn−1, I

hδcvh,τ

n−1)m + ∥η(tn)∥2m,

I2(tn) := 2∫

]tn−1,tn]

d∑i=0

(uh,τn , δh,iF i

t)mdt,

I3(tn) := −

∣∣∣∣∣∣∣∣∣∣∣∣∣∣τ (Lh

tn + Ihδ

)vh,τ

n +

d∑i=0

∫[tn−1,tn]

δh,iF itdt

∣∣∣∣∣∣∣∣∣∣∣∣∣∣2

m

+∣∣∣∣∣∣Ihδcvh,τ

n−1

∣∣∣∣∣∣2mτ2,

I4(tn) := 2∫

]tn−1,tn]

(vh,τ

n−1, 1n>1Nϱ;htn−1

vh,τn−1 +Gϱ

t

)m

dwϱt ,

I5(tn) := 2∫

]tn−1,tn]

∫Rd

(vh,τ

n−1, 1n>1Ih(z)vh,τ

n−1 + Rt(z))

mq(dt, dz),

I6(tn) :=(τIh

δcvh,τn−1, η(tn)

)m,

and where

η(tn) :=∫

]tn−1,tn]

(1n>1N

ϱ;htn−1

vh,τn−1 +Gϱ

t

)dwϱ

t +

∫]tn−1,tn]

∫Rd

(1n>1I


)q(dt, dz).

As in the proof Theorem 5.4.7, by Young’s inequality, (5.4.10), and (5.4.15), we have

E[∥vh,τ

n ∥2m

]≤

(1 + 2π1(|z| > δ)

)E

[∥φ∥2m

]− q2E

n∑l=1

τ

d∑i=1

∥δh,ivh,τl ∥

2m

+ En∑

l=1

τ(N2 + 2π1(|z| > δ) + τπ1(|z| > δ)2

)∥vh,τ

l ∥2m

+ NE∫

]0,tn]

d∑i=0

∥F it∥

2mdt + ∥Gt∥

2m,ℓ2+


2(dz)

dt.

SetZ := N2 + 2π1(|z| > δ),

R := max

2π1(|z| > δ)2√Z2 + 4π1(|z| > δ2 − Z

,N2

T.

Assume T > R. Making use of (5.4.29) and applying discrete Gronwall’s lemma, we get

5.5. Proof of the main theorems 155

that there exist a constant N(d,m,K, κ,T, δ) such that

E[∥vh,τ

n ∥2m

]+ E

n∑l=1

τ

d∑i=0

∥δh,ivh,τl ∥

2m ≤ NE

[∥φ∥2m

]

+NE∫

]0,T ]

d∑i=0

∥F it∥

2m + ∥Gt∥

2m,ℓ2+


2(dz)

dt. (5.4.30)

Using (5.4.15) instead of (5.4.16), we obtain (5.4.22) from (5.4.30) in the same manner asTheorem 5.4.7. Note that no bound on τ/h2 is needed in this case.

5.5 Proof of the main theorems

Proof of Theorem 5.2.3. By virtue of Theorems 2.9, 2.10, and 4.1 in [Gyö82], in order toobtain the existence, uniqueness, regularity, and the estimate (5.2.4), we only need to showthat (5.1.1) may be realized as an abstract stochastic evolution equation in a Gelfand tripleand that the growth condition and coercivity condition are satisfied. Indeed, since (5.1.1)is a linear equation, the hemicontinuity condition is immediate and monotonicity followsdirectly from the coercivity condition. By Holder’s inequality and Assumption 5.2.1(i),for u, v ∈ H1, we have

d∑i, j=0

(∂ ju, (v∂−ia

i jt + ai j∂−iv)

)0+

∫|z|>δ

u(· + z) − u − 1[−1,1](z)d∑

j=1

z j∂ ju, v

0

π1(dz)

+

∫|z|≤δ

∫ 1

0

d∑i, j=1

(z j∂ ju(· + θz), zi∂−iv

)0

(1 − θ)dθπ1(dz) ≤ N∥u∥1∥v∥1.

Therefore, since the pairing ⟨·, ·⟩1 brings (H1)∗ and H−1 into isomorphism, for each (ω, t) ∈[0,T ] ×Ω, there exists a linear operator At : H1 → H−1 such that ⟨v, Atu⟩1 agrees with theleft-hand-side of the above inequality and for u, v ∈ H1, ∥Atu∥−1 ≤ N∥u∥1. By Assumption5.2.2, the operator A defined by A(u) = Au+ f , maps H1 to H−1 and for u ∈ H1, ∥At(u)∥−1 ≤

N(∥u∥1 + ∥ f ∥−1).For an integer m ≥ 1, with abuse of notation, we write

(·, ·)m = ((1 − ∆)m/2·, (1 − ∆)m/2·)0.

and ∥ · ∥m for the corresponding norm in Hm. It is well known that the above inner productand norm are equivalent to the ones introduced in Section 1. For each m ≥ 1 and forall u ∈ Hm+1 and v ∈ Hm, we have (u, v)m ≤ ∥u∥m+1∥v∥m−1. Since Hm+1 is dense in Hm−1,we may define the pairing [·, ·]m : Hm+1 × Hm−1 → R by [v, v′]m = limn→∞(v, vn)m for all


v ∈ Hm+1 and v′ ∈ Hm−1, where (vn)∞n=1 ⊂ Hm+1 is such that ∥vn − v′∥m−1 → 0 as n→ ∞. Itcan be shown that the mapping from Hm−1 to (Hm+1)∗ given by v′ 7→ [·, v′]m is an isometricisomorphism. For more details, see [Roz90]. Therefore, for all m ≥ 0, (Hm+1,Hm,Hm−1)forms a Gelfand triple with the pairing [·, ·]m, where we make the convention that ⟨·, ·⟩1 =[·, ·]0.

For m ≥ 1 and all u ∈ Hm+1 and v ∈ Hm, using integration by parts, we get ⟨v, At(u)⟩1 =((Lt + It)u + f , v)0 = ⟨v, (Lt + It)u+ f ⟩1. Since this is true for all v ∈ Hm, which is dense inH1, the restriction of A to Hm+1 coincides with L+ I + f . Moreover, it can easily be shownunder Assumptions 5.2.1(i) and 5.2.2 that for all m ≥ 1 and u, v ∈ Hm+1, ∥At(u)∥m−1 ≤

N∥u∥m+1 + ∥ f ∥m−1, where N is a constant depending only on m, d,K, and ν, which showsthat A satisfies the growth condition. For u ∈ Hm, m ≥ 1, define Bϱ

t (u) = biϱt ∂iu + gϱt ,

Bt = (Bϱt )∞ϱ=1, and Cz(u) = u(· + z) − u + ot(z), z ∈ Rd. Owing to Assumption 5.2.1

(i), Bt is an operator from Hm+1 to Hm(ℓ2). Furthermore, C is an operator from Hm+1

to L2(Rd, π2(dz); Hm) (see (5.5.2)). It is also clear that A, B, and C are appropriatelymeasurable. Thus, (5.1.1) may be realized as the following stochastic evolution equationin the Gelfand triple (Hm+1,Hm,Hm−1):

ut = u0 +

∫]0,t]

As(us)ds +∫

]0,t]Bϱ

s(us)dwϱs +

∫]0,t]Cz(us−)q(dz, ds), (5.5.1)

for t ∈ [0,T ]. Let u ∈ C∞c . A simple calculation shows that there is a constant N = N(δ)such that ∫

Rd∥u(· + z) − u∥2mπ

2(dz) ≤ ς2(δ)∥u∥2m+1 + N∥u∥2m. (5.5.2)

Applying Holder’s inequality and the identity (u, ∂ ju) = 0, we obtain

∫|z|>δ′

u(· + z) − u − 1[−1,1](z)d∑

j=1

z j∂ ju, u

m

π1(dz) ≤ 0.

By the Holder’s inequality and the Cauchy-Bunyakovsky-Schwarz inequality, we have

2∫|z|≤δ′

∫ 1

0

d∑i, j=1

(z j∂ ju(· + θz), zi∂−iu

)m

(1 − θ)dθπ1(dz) ≤ ς1(δ)∥u∥2m+1.

There exists a constant ϵ = ϵ(κ, δ) such that

q := κ − ς(δ) − ϵ > 0.


As in Theorem 4.1.2 in [Roz90] and Lemma 5.4.4, using Holder’s and Young’s inequal-ities, the above estimates, and Assumption 5.2.1, we find that for each ϵ > 0, there is aconstant N = N(d,m, κ,K,T, δ) such that for all (ω, t) ∈ Ω × [0,T ],

2[u, At(u)]m + ∥Bt(u)∥2m,l2 +∫

Rd∥Cz(u)∥2mπ

2(dz) + q∥u∥2m+1

≤ N(∥u∥2m + ∥ ft∥m−1 + ∥gt∥m,ℓ2 +

∫Rd∥ot(z)∥2mπ

2(dz)).

Using the self-adjointness of (1 − ∆)1/2, the properties of the CBF [·, ·]m, and Assumption5.2.2, for all v ∈ C∞c and u ∈ Hm+1, m ≥ 1, we have

[v, A(u)]m = ((L + I)u, (1 − ∆)mv)0 + ( f , (1 − ∆)mv)0. (5.5.3)

Owing to (5.5.3) and the denseness of (1 − ∆)−mC∞c in H1, from Theorems 2.9, 2.10, and4.1 in [Gyö82], we obtain the existence and uniqueness of a solution u of (5.1.1), such thatu is a càdlàg Hm-valued process satisfying (5.2.4).

Proof of Proposition 5.2.5. Let A, B, and C be as in (5.5.1). Owing to Assumption 5.2.1,the boundedness of the m−1-norm of g in expectation, and estimate (5.2.4), using Jensen’sinequality and Itô’s isometry, for s, t ∈ [0,T ], we get

E∣∣∣∣∣∣∣∣∣∣∣∣∫

]s,t]Ar(ur)ds

∣∣∣∣∣∣∣∣∣∣∣∣2m−1

≤ |t − s|(NE

∫]0,T ]∥ut∥

2m+1dt + E

∫]0,T ]∥ fr∥

2m−1dr

)≤ N |t − s|,

E∣∣∣∣∣∣∣∣∣∣∣∣∫

]s,t]Bϱ

r (ur)dwρr

∣∣∣∣∣∣∣∣∣∣∣∣2m−1

= E∫

]s,t]∥Br(ur)∥2m−1,ℓ2

dr

≤ N |t − s|(supt≤T

E∥ut∥2m + sup

t≤TE∥gt∥m−1,ℓ2

)≤ N |t − s|,

and

E∣∣∣∣∣∣∣∣∣∣∣∣∫

]s,t]

∫RdCz(ur−)q(dr, dz)

∣∣∣∣∣∣∣∣∣∣∣∣2m−1

= E∫

]s,t]

∫Rd∥Cz(ur)∥2m−1π

2(dz)ds

≤ N |t − s|(supt≤T

E∥ut∥2m + sup

t≤TE


2(dz))≤ N |t − s|,

which completes the proof of the proposition.

Theorem 5.5.1. Let Assumptions 5.2.1 through 5.2.5 hold for some m ≥ 2. Let u bethe solution of (5.1.1) and (uh,τ

n )Tn=0 be defined by (5.4.1). Then there is a constant N =N(d,m, κ,K,T,C, λ, κ2

m, δ) such that

E[

max0≤n≤T

∥utn − uh,τn ∥

2m−2

]+ E

T∑l=0

τ

d∑i=0

∥δh,iutl − δh,iuh,τl ∥

2m−2ds ≤ N(τ + |h|2). (5.5.4)


Proof. For t ∈ [0,T ], let κ1(t) := tn−1 for t ∈]tn−1, tn], and set eh,τn := uh,τ

n − utn . One caneasily verify that eh,τ

n satisfies in Hm−2,

eh,τn = eh,τ

n−1 +

∫]tn−1,tn]

(Lhtn−1 + Ih)eh,τ

n−1 +

d∑i=0

δh,iF it

dt +∫

]tn−1,tn]

(N

ϱ;htn−1

eh,τn−1 +Gϱ

t

)dwϱ

t

+

∫]tn−1,tn]

∫Rd

(Ih(z)eh,τ

n−1 + Rt(z))

q(dt, dz),

where

F0t := (Lh

κ1(t) − Lκ1(t))ut + (Lκ1(t) − Lt)ut + (Ih − I)ut + ( fκ1(t) − ft) + Ihδc(uκ1(t) − ut)

+

d∑j=1

a0 jκ1(t)δ−h, j(uκ1(t) − ut) +

d∑i=0

ai0κ1(t)δh,i(uκ1(t) − ut) −

d∑i, j=1

δ−h, j(uκ1(t) − ut)(· + h)δh,iai jκ1(t),

F it :=

d∑j=1

ai jκ1(t)δ−h, j(uκ1(t) − ut) +

∞∑k=0

χ(h,k)∑l=1

θk,hl ζ

i jk,hδ−h, j(uκ1(t) − ut)(· + hzrh,k

l)

Gϱt : = (Nϱ

κ1(t) − Nϱt )ut + (Nϱ;h

κ1(t) − Nϱκ1(t))ut +N

ϱκ1(t)(uκ1(t) − ut) + (gϱκ1(t) − gϱt )

Rht (z) : =

(Ih(z) − I(z)

)ut− + I

h(z)(uκ1(t) − ut−) +(oκ1(t)(z) − ot(z)

).

By Theorem 5.4.7, we have

E[

max0≤n≤T

∥eh,τn ∥

2m−2

]+ E

M∑n=0

τ

d∑i=0

∥δh,ieh,τn ∥

2m−2

≤ NE∫

]0,T ]

( d∑i=0

∥F it∥

2m−2 + ∥Gt∥

2m−2,ℓ2

+

∫Rd∥Rt(z)∥2m−2π

2(dz))dt.

Using Lemmas 5.4.1, 5.4.2, and 5.4.3 and Assumptions 5.2.1(i) and 5.2.4, the right-hand-side of the above relation can be estimated by

NE∫

]0,T ]

(|h|2∥ut∥

2m+1 + |κ1(t) − t|∥ut∥

2m + ∥uκ1(t) − ut∥

2m−1

)dt

+NE∫

]0,T ]

(∥ fκ1(t) − ft∥

2m−2 + ∥gκ1(t) − gt∥m−2,ℓ2 +

∫Rd∥oκ1(t)(z) − ot(z)∥m−2π

2(dz))

dt

where N depends only on d,m, κ,K,C, λ,T, δ and ν. By virtue of (5.2.4), Proposition


5.2.5, and Assumption 5.2.3, we obtain (5.5.4), which completes the proof.

Theorem 5.5.2. Let Assumptions 5.2.1 through 5.2.4 hold with m ≥ 2 and let u be thesolution of (5.1.1). There exists a constant R = R(d,m, κ,K, δ) such that if T > R, thenthere exists a unique solution (vh,τ)Tn=0 of (5.4.2) in Hm−2. Moreover, there is a constantN = N(d,m, κ,K,T,C, λ, κ2

m, δ) such that

E[

max0≤n≤T

∥utn − vh,τn ∥

2m−2

]+ E

T∑l=0

τ

d∑i=0

∥δh,iutl − δh,ivh,τl ∥

2m−2ds ≤ N(τ + |h|2). (5.5.5)

Proof. The existence and uniqueness follows directly from Theorem 5.4.8. Let κ1(t) beas in the previous proof and set κ2(t) = tn for t ∈]tn−1, tn]. Let G and R be defined as inTheorem 5.5.1 and define F i to be F i with κ1(t) replaced with κ2(t). Set eh,τ

n = vh,τn − utn . As

in the proof of Theorem 5.5.1, we have

eh,τn = eh,τ

n−1 +

∫]tn−1,tn]

(Lhtn + Ih

δ )eh,τn + Ih

δceh,τn−1 +

d∑i=0

δh,iF it

dt

+

∫]tn−1,tn]

(1n>1N

ϱ;htn−1

eh,τn−1 + Gϱ

t

)dwϱ

t +

∫]tn−1,tn]

∫Rd

(1n>1I

h(z)eh,τn−1 + Rt(z)

)q(dt, dz),

where

F i = F i, for i , 0, F0 = F0 + Ihδc(uκ1(t) − uκ2(t)),

Gϱt = 1t≤t1(N

ϱt ut + gϱt ) + 1t>t1G

ϱt , Rt(z) = 1t≤t1I(z)ut− + 1t>t1Rt(z).

By Theorem 5.4.8, we have

E[

max0≤n≤T

∥eh,τn ∥

2m−2

]+ E

M∑n=0

τ

d∑i=0

∥δh,ieh,τn ∥

2m−2 ≤ N(A1 + A2 + A3),

where

A1 := E∫

]0,T ]

d∑i=0

∥F it∥

2m−2dt +

∫]t1,T ]

(∥Gt∥

2m−2,ℓ2

+

∫Rd∥Rt(z)∥2m−2π

2(dz))

dt,

A2 := E∫

]0,T ]∥Ihδc(uκ1(t) − uκ2(t))∥2m−2dt

A3 := E∫

]0,t1]

(∥Mtut + gt∥

2m−2,ℓ2

+

∫Rd∥I(z)ut + ot(z)∥2m−2π

2(dz))

dt.


As in the proof of Theorem 5.5.1, we have A1 ≤ N(τ + |h|2). By Proposition 5.2.5, we get

A2 ≤ NE∫ T

0∥uκ1(t) − uκ2(t)∥

2m−1dt ≤ Nτ.

Owing to (5.2.3), we have

A3 ≤ NE∫ t1

0

(∥ut∥

2m−1 + ∥gt∥

2m−2,ℓ2

+


2(dz))

dt

≤ NτE∫ t1

0

(supt≤T∥ut∥

2m−1 + ξ

)dt ≤ Nτ.

Combining the above estimates yields (5.5.5).

By virtue of Sobolev’s embedding theorem and (5.2.10), as in [GK10], we obtain thefollowing corollaries of Theorem 5.5.1 and Theorem 5.5.2.

Corollary 5.5.3. Suppose the assumptions of Theorem 5.5.1 hold with m > n + 2 + d/2,where n is an integer with n ≥ 0. Then for all λ = (λ1, . . . , λn) ∈ 1 . . . , dn and δh,λ =

δh,λ1 · · · δh,λn , there is a constant N = N(d,m, κ,K,T,C, λ, κ2m, δ) such that

E[

max0≤n≤T

supx∈Rd|δh,λutn(x) − δh,λuh,τ

n (x)|2]+ E

[max

0≤n≤T∥δh,λutn − δh,λuh,τ

n ∥2ℓ2(Gh)

]≤ N(τ + |h|2).

Corollary 5.5.4. Suppose the assumptions of Theorem 5.5.2 hold with m > n + 2 + d/2,where n is an integer with n ≥ 0. Then for all λ = (λ1, . . . , λn) ∈ 1 . . . , dn and δh,λ =

δh,λ1 · · · δh,λn , there is a constant N = N(d,m, κ,K,T,C, λ, κ2m, δ) such that

E[

max0≤n≤T

supx∈Rd|δh,λutn(x) − δh,λvh,τ

n (x)|2]+ E

[max

0≤n≤T∥δh,λutn − δh,λvh,τ

n ∥2ℓ2(Gh)

]≤ N(τ + |h|2).

Proof of Theorems 5.2.8 and 5.2.9. Let (uh,τn )M

n=0 be defined by (5.2.8). Denote by (·, ·)ℓ2(Gh)

the inner product of ℓ2(Gh). There exists a constant ϵ = ϵ(κ, δ) such that

q := κ − ς1(δ) − ϵ > 0.

As in (5.4.11), there is a constant N6 = N6(d, κ,K, δ) such that for all ϕ ∈ ℓ2(Gh),

(ϕ, Lht ϕ)ℓ2(Gh) + (ϕ, Ih

δϕ)ℓ2(Gh) ≤ −qd∑

i=1

∥δh,iϕ∥2ℓ2(Gh) + N6∥ϕ∥

2ℓ2(Gh).

Following the arguments in the beginning of the proof of Theorem 5.4.8, we conclude thatif T > N6T , then there exists a unique solution (vh,τ

n )Mn=0 in ℓ2(Gh) of (5.2.9). It is easy

to see that N6 < N2 (for the same choice of ϵ) for all m > 0, where N2 is the constant


appearing on the right-hand-side of (5.4.11), and hence N6 < R, where R is as in Theorem5.4.8. Let (uh,τ

n )Mn=1 be defined by (5.4.1). By Theorem 5.5.2, there exists a unique solution

(vh,τn )M

n=1 of (5.4.2). It suffices to show that almost surely,

uh,τn (x) = uh,τ

n (x) (5.5.6)

andvh,τ

n (x) = vh,τn (x), (5.5.7)

for all n ∈ 0, ...,M and x ∈ Gh. Let S : Hm−2 → ℓ2(Gh) denote the embedding fromRemark 5.2.6. Applying S to both sides of (5.4.1), one can see that S uh,τ and uh,τ satisfythe same recursive relation in ℓ2(Gh) with common initial condition φ, and hence (5.5.6)follows. Similarly, S vh,τ and vh,τ satisfy the same equation in ℓ2(Gh) and (5.5.7) followsfrom the uniqueness of the ℓ2(Gh) solution of (5.2.9).

Remark 5.5.5. It follows from Corollaries 5.5.3, 5.5.4, and relations (5.5.6) and (5.5.7)that if more regularity is assumed of the coefficients and the data of the equation (5.1.1),then better estimates can be obtained than the ones presented in Theorems 5.2.8 and 5.2.9.

Chapter 6Conclusions and future workIn this thesis, we investigated existence, uniqueness, and regularity of degenerate paraboliclinear SIDEs in the whole space with adapted coefficients from a couple of standpoints.First, we used the method of stochastic characteristics to derive classical solutions directly.Second, we derived the integer scale L2-Sobolev theory for the equations using the varia-tional framework of stochastic evolution equations and the method of vanishing viscosity.We then established the rate of convergence of some finite difference schemes for a simpleclass of SIDEs under the assumption of non-degenerate stochastic parabolicity.

There are myriad of future directions to consider. We will only discuss a few. Aswe mentioned in Chapter 4, the integer scale Lp-Sobolev theory for degenerate SIDEs iscurrently underway and will be available soon. Still though, it would be ideal to obtainan existence and regularity theory in the full scale of Bessel potential spaces with Hölderassumptions on the coefficients. It is also interesting to study non-degenerate linear SIDEsunder weaker regularity conditions than those required for the fully degenerate theory.Currently, an existence theory is known only in a few special cases; we refer the reader to[MP09, MP11, MP12, MP13, KK12b, KK12a, KKK13] for some relatively recent resultsin this direction. With regards to approximations of SIDEs, we considered only equationswith non-degenerate stochastic parabolicity and where the integral operators do not dependon the space variable. It would be interesting to relax these conditions in the future. Also,from a practical standpoint, the error from truncating the domain should be studied in con-junction with the numerical approximations of the equations. It is worth mentioning thata regularity theory for SIDEs in bounded domains with degenerate stochastic parabolicityis non-existent as far as the author knows. There is also no reason to confine oneself tofinite difference schemes, since there are other numerical methods such as finite elementsand wavelets. Although no simulations were done in this thesis, they will be considered ina future work along with some applications to the non-linear filtering problem.

162

References[Ber77] Melvin S. Berger. Nonlinearity and functional analysis. Academic Press

[Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Lec-tures on nonlinear problems in mathematical analysis, Pure and AppliedMathematics.

[Bis81] Jean-Michel Bismut. Mécanique Aléatoire, volume 866 of Lecture Notes inMathematics. Springer-Verlag, Berlin-New York, 1981. With an Englishsummary.

[BL12] Andrea Barth and Annika Lang. Milstein approximation for advection-diffusion equations driven by multiplicative noncontinuous martingalenoises. Appl. Math. Optim., 66(3):387–413, 2012.

[BvNVW08] Z. Brzezniak, J. M. A. M. van Neerven, M. C. Veraar, and L. Weis. Itô’sformula in UMD Banach spaces and regularity of solutions of the Zakaiequation. J. Differential Equations, 245(1):30–58, 2008.

[CJM92] P. L. Chow, J.-L. Jiang, and J.-L. Menaldi. Pathwise convergence of ap-proximate solutions to Zakai’s equation in a bounded domain. In Stochasticpartial differential equations and applications (Trento, 1990), volume 268of Pitman Res. Notes Math. Ser., pages 111–123. Longman Sci. Tech., Har-low, 1992.

[CK10] Zhen-Qing Chen and Kyeong-Hun Kim. An Lp-Theory of Non-DivergenceForm SPDEs Driven by Lévy Processes. arXiv preprint arXiv:1007.3295,2010.

[CV05] Rama Cont and Ekaterina Voltchkova. A finite difference scheme for optionpricing in jump diffusion and exponential Lévy models. SIAM J. Numer.Anal., 43(4):1596–1626 (electronic), 2005.

[DMGZ94] Giuseppe De Marco, Gianluca Gorni, and Gaetano Zampieri. Global inver-sion of functions: an introduction. NoDEA Nonlinear Differential EquationsAppl., 1(3):229–248, 1994.

[DPMT07] Giuseppe Da Prato, Jose-Luis Menaldi, and Luciano Tubaro. Some resultsof backward Itô formula. Stoch. Anal. Appl., 25(3):679–703, 2007.

[DPT98] Giuseppe Da Prato and Luciano Tubaro. Some remarks about backward Itôformula and applications. Stochastic Anal. Appl., 16(6):993–1003, 1998.

[DPZ92] Giuseppe Da Prato and Jerzy Zabczyk. Stochastic Equations in Infinite Di-mensions, volume 44 of Encyclopedia of Mathematics and its Applications.Cambridge University Press, Cambridge, 1992.

[FGP10] F. Flandoli, M. Gubinelli, and E. Priola. Well-posedness of the transportequation by stochastic perturbation. Invent. Math., 180(1):1–53, 2010.

163

[FK85] Tsukasa Fujiwara and Hiroshi Kunita. Stochastic differential equations ofjump type and Lévy processes in diffeomorphisms group. J. Math. KyotoUniv., 25(1):71–106, 1985.

[GGK14] Máté Gerencsér, István Gyöngy, and N. V. Krylov. On the solvability ofdegenerate stochastic partial differential equations in Sobolev spaces. arXivpreprint arXiv:1404.4401, 2014.

[GK81] István Gyöngy and N. V. Krylov. On stochastics equations with respect tosemimartingales. II. Itô formula in Banach spaces. Stochastics, 6(3-4):153–173, 1981.

[GK92] István Gyöngy and Nicolai V. Krylov. On stochastic partial differentialequations with unbounded coefficients. Potential Anal., 1(3):233–256,1992.

[GK96] W. Grecksch and P. E. Kloeden. Time-discretised Galerkin approximationsof parabolic stochastic PDEs. Bull. Austral. Math. Soc., 54(1):79–85, 1996.

[GK10] István Gyöngy and Nicolai Krylov. Accelerated finite difference schemesfor linear stochastic partial differential equations in the whole space. SIAMJ. Math. Anal., 42(5):2275–2296, 2010.

[GK11] István Gyöngy and Nicolai Krylov. Accelerated numerical schemes forPDEs and SPDEs. In Stochastic analysis 2010, pages 131–168. Springer,Heidelberg, 2011.

[GK81] István Gyöngy and N. V. Krylov. On stochastic equations with respect tosemimartingales. I. Stochastics, 4(1):1–21, 1980/81.

[GM83] B. Grigelionis and R. Mikulevicius. Stochastic evolution equations and den-sities of the conditional distributions. In Theory and Application of RandomFields, volume 49 of Lecture Notes in Control and Inform. Sci., pages 49–88. Springer, Berlin, 1983.

[GM05] István Gyöngy and Annie Millet. On discretization schemes for stochasticevolution equations. Potential Anal., 23(2):99–134, 2005.

[GM09] István Gyöngy and Annie Millet. Rate of convergence of space time approx-imations for stochastic evolution equations. Potential Anal., 30(1):29–64,2009.

[GM11] B. Grigelionis and R. Mikulevicius. Nonlinear filtering equations forstochastic processes with jumps. In The Oxford Handbook of NonlinearFiltering, pages 95–128. Oxford Univ. Press, Oxford, 2011.

[Gri72] B. I. Grigelionis. Stochastic equations for the nonlinear filtering of randomprocesses. Litovsk. Mat. Sb., 12(4):37–51, 233, 1972.

[Gri76] B. Grigelionis. Reduced stochastic equations of nonlinear filtering of ran-dom processes. Litovsk. Mat. Sb., 16(3):51–63, 232, 1976.

164

[Gri82] B. Grigelionis. Stochastic nonlinear filtering equations and semimartin-gales. In Nonlinear filtering and stochastic control (Cortona, 1981), vol-ume 972 of Lecture Notes in Math., pages 63–99. Springer, Berlin-NewYork, 1982.

[GT01] David Gilbarg and Neil S. Trudinger. Elliptic Partial Differential Equationsof Second Order. Classics in Mathematics. Springer-Verlag, Berlin, 2001.Reprint of the 1998 edition.

[Gyö82] I. Gyöngy. On stochastic equations with respect to semimartingales. III.Stochastics, 7(4):231–254, 1982.

[Gyö98] István Gyöngy. Lattice approximations for stochastic quasi-linear parabolicpartial differential equations driven by space-time white noise. I. PotentialAnal., 9(1):1–25, 1998.

[Gyö99] István Gyöngy. Lattice approximations for stochastic quasi-linear parabolicpartial differential equations driven by space-time white noise. II. PotentialAnal., 11(1):1–37, 1999.

[Gyö11] I. Gyöngy. On finite difference schemes for degenerate stochastic parabolicpartial differential equations. J. Math. Sci. (N. Y.), 179(1):100–126, 2011.Problems in mathematical analysis. No. 61.

[Hal12] Eric Joseph Hall. Accelerated spatial approximations for time discretizedstochastic partial differential equations. SIAM J. Math. Anal., 44(5):3162–3185, 2012.

[Hal13] Eric Joseph Hall. Higher order spatial approximations for degenerateparabolic stochastic partial differential equations. SIAM J. Math. Anal.,45(4):2071–2098, 2013.

[Hau08] Erika Hausenblas. Finite element approximation of stochastic partial differ-ential equations driven by Poisson random measures of jump type. SIAM J.Numer. Anal., 46(1):437–471, 2007/08.

[Hau05] Erika Hausenblas. Existence, uniqueness and regularity of parabolic SPDEsdriven by Poisson random measure. Electron. J. Probab., 10:1496–1546,2005.

[HM06] Erika Hausenblas and Iuliana Marchis. A numerical approximation ofparabolic stochastic partial differential equations driven by a Poisson ran-dom measure. BIT, 46(4):773–811, 2006.

[HØUZ10] Helge Holden, Bernt Øksendal, Jan Ubøe, and Tusheng Zhang. StochasticPartial Differential Equations: A Modeling, White Noise Functional Ap-proach. Universitext. Springer, New York, second edition, 2010.

[Jac79] Jean Jacod. Calcul Stochastique et Problèmes de Martingales, volume 714of Lecture Notes in Mathematics. Springer, Berlin, 1979.

[JK10] Arnulf Jentzen and Peter Kloeden. Taylor expansions of solutions ofstochastic partial differential equations with additive noise. Ann. Probab.,38(2):532–569, 2010.

165

[JS03] Jean Jacod and Albert N. Shiryaev. Limit Theorems for Stochastic Pro-cesses, volume 288 of Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences]. Springer-Verlag,Berlin, second edition, 2003.

[Kal97] Olav Kallenberg. Foundations of Modern Probability. Probability and itsApplications (New York). Springer-Verlag, New York, 1997.

[KK12a] Ildoo Kim and Kyeong-Hun Kim. A generalization of the Littlewood-Paley inequality for the fractional Laplacian (−∆)α/2. J. Math. Anal. Appl.,388(1):175–190, 2012.

[KK12b] Kyeong-Hun Kim and Panki Kim. An Lp-theory of a class of stochasticequations with the random fractional Laplacian driven by Lévy processes.Stochastic Process. Appl., 122(12):3921–3952, 2012.

[KKK13] Ildoo Kim, Kyeong-Hun Kim, and Panki Kim. Parabolic Littlewood-Paleyinequality for ϕ(−∆)-type operators and applications to stochastic integro-differential equations. Adv. Math., 249:161–203, 2013.

[KM11] Reiichiro Kawai and Hiroki Masuda. On simulation of tempered stablerandom variates. J. Comput. Appl. Math., 235(8):2873–2887, 2011.

[Kom84] Takashi Komatsu. On the martingale problem for generators of stable pro-cesses with perturbations. Osaka J. Math., 21(1):113–132, 1984.

[KR77] N. V. Krylov and B. L. Rozovskiı. The Cauchy problem for linear stochasticpartial differential equations. Izv. Akad. Nauk SSSR Ser. Mat., 41(6):1329–1347, 1448, 1977.

[KR79] N. V. Krylov and B. L. Rozovskiı. Stochastic evolution equations. In Cur-rent Problems in Mathematics, Vol. 14 (Russian), pages 71–147, 256. Akad.Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979.

[KR81] N. V. Krylov and B. L. Rozovskiı. On the first integrals and Liouville equa-tions for diffusion processes. In Stochastic differential systems (Visegrád,1980), volume 36 of Lecture Notes in Control and Information Sci., pages117–125. Springer, Berlin-New York, 1981.

[KR82] N. V. Krylov and B. L. Rozovskiı. Characteristics of second-order degener-ate parabolic Itô equations. Trudy Sem. Petrovsk., 8:153–168, 1982.

[Kry99] N. V. Krylov. An analytic approach to SPDEs. In Stochastic Partial Differ-ential Equations: Six Perspectives, volume 64 of Math. Surveys Monogr.,pages 185–242. Amer. Math. Soc., Providence, RI, 1999.

[Kry08] N. V. Krylov. Lectures on Elliptic and Parabolic Equations in SobolevSpaces, volume 96 of Graduate Studies in Mathematics. American Mathe-matical Society, Providence, RI, 2008.

[Kry11] N. V. Krylov. On the Itô-Wentzell formula for distribution-valued processesand related topics. Probab. Theory Related Fields, 150(1-2):295–319, 2011.

166

[Kun81] Hiroshi Kunita. On the decomposition of solutions of stochastic differentialequations. In Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham,1980), volume 851 of Lecture Notes in Math., pages 213–255. Springer,Berlin-New York, 1981.

[Kun86a] H. Kunita. Lectures on Stochastic Flows and Applications, volume 78of Tata Institute of Fundamental Research Lectures on Mathematics andPhysics. Published for the Tata Institute of Fundamental Research, Bom-bay; by Springer-Verlag, Berlin, 1986.

[Kun86b] Hiroshi Kunita. Convergence of stochastic flows with jumps and Lévy pro-cesses in diffeomorphisms group. Ann. Inst. H. Poincaré Probab. Statist.,22(3):287–321, 1986.

[Kun96] Hiroshi Kunita. Stochastic differential equations with jumps and stochas-tic flows of diffeomorphisms. In Itô’s Stochastic Calculus and ProbabilityTheory, pages 197–211. Springer, Tokyo, 1996.

[Kun97] Hiroshi Kunita. Stochastic Flows and Stochastic Differential Equations,volume 24 of Cambridge Studies in Advanced Mathematics. CambridgeUniversity Press, Cambridge, 1997. Reprint of the 1990 original.

[Kun04] Hiroshi Kunita. Stochastic differential equations based on Lévy processesand stochastic flows of diffeomorphisms. In Real and Stochastic Analysis,Trends Math., pages 305–373. Birkhäuser Boston, Boston, MA, 2004.

[Lan12] Annika Lang. Almost sure convergence of a Galerkin approximation forSPDEs of Zakai type driven by square integrable martingales. J. Comput.Appl. Math., 236(7):1724–1732, 2012.

[Len77] E. Lenglart. Relation de domination entre deux processus. Ann. Inst. H.Poincaré Sect. B (N.S.), 13(2):171–179, 1977.

[LM14a] James-Michael Leahy and Remigijus Mikulevicius. On classical solu-tions of linear stochastic integro-differential equations. arXiv preprintarXiv:1404.0345, 2014.

[LM14b] James-Michael Leahy and Remigijus Mikulevicius. On degenerate linearstochastic evolution equations driven by jump processes. arXiv preprintarXiv:1406.4541, 2014.

[LM14c] James-Michael Leahy and Remigijus Mikulevicius. On some properties ofspace inverses of stochastic flows. arXiv preprint arXiv:1411.6277, 2014.

[LR04] Gabriel J. Lord and Jacques Rougemont. A numerical scheme for stochasticPDEs with Gevrey regularity. IMA J. Numer. Anal., 24(4):587–604, 2004.

[LS89] R. Sh. Liptser and A. N. Shiryayev. Theory of martingales, volume 49 ofMathematics and its Applications (Soviet Series). Kluwer Academic Pub-lishers Group, Dordrecht, 1989. Translated from the Russian by K. Dz-japaridze.

[MB07] Thilo Meyer-Brandis. Stochastic Feynman-Kac equations associated toLévy-Itô diffusions. Stoch. Anal. Appl., 25(5):913–932, 2007.

167

[Mey76] P. A. Meyer. La théorie de la prédiction de F. Knight. In Séminaire de Prob-abilités, X (Première partie, Univ. Strasbourg, Strasbourg, année universi-taire 1974/1975), pages 86–103. Lecture Notes in Math., Vol. 511. Springer,Berlin, 1976.

[Mey81] P.A. Meyer. Flot D’une Equation Differentielle Stochastique (d’aprèsMalliavin, Bismut, Kunita), volume 850, pages 103–117. Springer, Berlin-Heidelberg, 1981.

[Mik83] R. Mikulevicius. Properties of solutions of stochastic differential equations.Litovsk. Mat. Sb., 23(4):18–31, 1983.

[Mik00] R. Mikulevicius. On the Cauchy problem for parabolic SPDEs in Hölderclasses. Ann. Probab., 28(1):74–103, 2000.

[MP09] R. Mikulevicius and H. Pragarauskas. On Hölder solutions of the integro-differential Zakai equation. Stochastic Process. Appl., 119(10):3319–3355,2009.

[MP11] R. Mikulevicius and H. Pragarauskas. Model problem for integro-differential Zakai equation with discontinuous observation processes. Appl.Math. Optim., 64(1):37–69, 2011.

[MP12] R. Mikulevicius and H. Pragarauskas. On Lp-estimates of some singularintegrals related to jump processes. SIAM J. Math. Anal., 44(4):2305–2328,2012.

[MP13] R. Mikulevicius and H. Pragarauskas. On Lp- theory for stochastic parabolicintegro-differential equations. Stoch PDE: Anal Comp, 1(2):282–324, 2013.

[MP76] Michel Métivier and Giovanni Pistone. Une formule d’isométrie pourl’intégrale stochastique hilbertienne et équations d’évolution linéairesstochastiques. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(1):1–18, 1975/76.

[MR99] R. Mikulevicius and B. L. Rozovskii. Martingale problems for stochas-tic PDE’s. In Stochastic partial differential equations: six perspectives,volume 64 of Math. Surveys Monogr., pages 243–325. Amer. Math. Soc.,Providence, RI, 1999.

[Nov75] A. A. Novikov. Discontinuous martingales. Teor. Verojatnost. i Primemen.,20:13–28, 1975.

[Ole65] O. A. Oleınik. On the smoothness of solutions of degenerating elliptic andparabolic equations. Dokl. Akad. Nauk SSSR, 163:577–580, 1965.

[OP89] Daniel Ocone and Étienne Pardoux. A generalized Itô-Ventzell formula.Application to a class of anticipating stochastic differential equations. Ann.Inst. H. Poincaré Probab. Statist., 25(1):39–71, 1989.

[OR71] O. A. Oleınik and E. V. Radkevic. Second order equations with non-negative characteristic form. In Mathematical Analysis, 1969 (Russian),pages 7–252. Akad. Nauk SSSR Vsesojuzn. Inst. Naucn. i Tehn. Informa-cii, Moscow, 1971.

168

[Par72] Étienne Pardoux. Sur des équations aux dérivées partielles stochastiquesmonotones. C. R. Acad. Sci. Paris Sér. A-B, 275:A101–A103, 1972.

[Par75] Étienne Pardoux. Équations aux dérivées partielles stochastiques de typemonotone. In Séminaire sur les Équations aux Dérivées Partielles (1974–1975), III, Exp. No. 2, page 10. Collège de France, Paris, 1975.

[Pri12] Enrico Priola. Pathwise uniqueness for singular SDEs driven by stable pro-cesses. Osaka J. Math., 49(2):421–447, 2012.

[Pri14] Enrico Priola. Stochastic flow for SDEs with jumps and irregular drift term.arXiv preprint arXiv:1405.2575, 2014.

[Pro05] Philip E. Protter. Stochastic integration and differential equations, vol-ume 21 of Stochastic Modelling and Applied Probability. Springer-Verlag,Berlin, 2005. Second edition. Version 2.1, Corrected third printing.

[PV14] Matthijs Pronk and Mark Veraar. A new approach to stochastic evolutionequations with adapted drift. J. Differential Equations, 256(11):3634–3683,2014.

[PZ07] S. Peszat and J. Zabczyk. Stochastic Partial Differential Equations withLévy Noise: An Evolution Equation Approach, volume 113 of Encyclo-pedia of Mathematics and its Applications. Cambridge University Press,Cambridge, 2007.

[PZ13] S. Peszat and J. Zabczyk. Time regularity of solutions to linear equa-tions with Lévy noise in infinite dimensions. Stochastic Process. Appl.,123(3):719–751, 2013.

[QZ08] Huijie Qiao and Xicheng Zhang. Homeomorphism flows for non-Lipschitzstochastic differential equations with jumps. Stochastic Process. Appl.,118(12):2254–2268, 2008.

[Roz90] B. L. Rozovskiı. Stochastic Evolution Systems: Linear Theory and Appli-cations to Nonlinear Filtering, volume 35 of Mathematics and its Applica-tions (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1990.Translated from the Russian by A. Yarkho.

[RZ07] Michael Röckner and Tusheng Zhang. Stochastic evolution equations ofjump type: existence, uniqueness and large deviation principles. PotentialAnal., 26(3):255–279, 2007.

[Tin77a] E. Tinfavicius. Linearized stochastic equations of nonlinear filtering of ran-dom processes. Lithuanian Mathematical Journal, 17(3):321–334, 1977-07-01 1977.

[Tin77b] E. Tinfavicius. On the reduced stochastic equations for non-linear filteringof random processes. Litovsk. Mat. Sb., 17(3):53–72, 212, 1977.

[Tri10] Hans Triebel. Theory of function spaces. Modern Birkhäuser Classics.Birkhäuser/Springer Basel AG, Basel, 2010. Reprint of 1983 edition[MR0730762], Also published in 1983 by Birkhäuser Verlag [MR0781540].

169

[Ver12] Mark Veraar. The stochastic Fubini theorem revisited. Stochastics,84(4):543–551, 2012.

[Wal86] John B. Walsh. An Introduction to Stochastic Partial Differential Equations.In École d’été de Probabilités de Saint-Flour, XIV—1984, volume 1180 ofLecture Notes in Math., pages 265–439. Springer, Berlin, 1986.

[Yan05] Yubin Yan. Galerkin finite element methods for stochastic parabolic partialdifferential equations. SIAM J. Numer. Anal., 43(4):1363–1384 (electronic),2005.

[Yoo00] Hyek Yoo. Semi-discretization of stochastic partial differential equationson R1 by a finite-difference method. Math. Comp., 69(230):653–666, 2000.

[Zha13] Xicheng Zhang. Degenerate irregular SDEs with jumps and application tointegro-differential equations of Fokker-Planck type. Electron. J. Probab.,18:no. 55, 25, 2013.

[Zho13] Guoli Zhou. Global well-posedness of a class of stochastic equations withjumps. Adv. Difference Equ., page 2013:175, 2013.

170

on parabolic stochastic integro-differential …on parabolic stochastic integro-differential...

Documents