VISCOSITY SOLUTIONS OF INFINITEDIMENSIONAL BLACK-SCHOLES-BARENBLATT

EQUATION

Djivede Kelome and Andrzej Swiech ∗

School of Mathematics

Georgia Institute of TechnologyAtlanta, GA 30332

e-mails: kelome@math.gatech.eduswiech@math.gatech.edu

Abstract

We study an infinite dimensional Black-Scholes-Barenblatt equation which is aHamilton-Jacobi-Bellman equation that is related to option pricing in the Musielamodel of interest rate dynamics. We prove the existence and uniqueness of viscos-ity solutions of the Black-Scholes-Barenblatt equation and discuss their stochasticoptimal control interpretation. We also show that in some cases the solution can belocally uniformly approximated by solutions of suitable finite dimensional Hamilton-Jacobi-Bellman equations.

Key words: Viscosity solutions, Hamilton-Jacobi-Bellman equations, stochastic optimalcontrol, option pricing, forward rates.

2000 Mathematics Subject Classification: 49L25, 35R15, 93E20, 91B28.

1 Introduction

We investigate a class of Hamilton-Jacobi-Bellman (HJB) equations related to the Musielamodel of interest rates [30] that describes the dynamics of forward rates in terms ofevolution of infinite dimensional diffusion processes. In this model the forward rate processr(t, u)u,t≥0 evolves according to a stochastic differential equation

dr(t, u) =

(∂

∂ur(t, u) +

d∑i=1

τi(t, u)

∫ u

0

τi(t, µ)dµ

)dt+

d∑i=1

τi(t, u)dwit,

∗A. Swiech was supported by NSF grant DMS 0098565.

1

where W = (w1, ..., wd) is a standard d-dimensional Brownian motion, and τi are certainfunctions. Using the notation A = − d

du, τt(u) = (τ1(t, u), ..., τd(t, u)),

c(τt)(u) =

d∑i=1

τi(t, u)

∫ u

0

τi(t, µ)dµ,

the above equation can be written as an abstract infinite dimensional stochastic differentialequation

drt = (−Art + c(τt))dt+ τt · dWt, r0 ∈ H, (1)

where H is some separable Hilbert space of functions on R+ (for instance H = L2(R+),

H = H1(R+) or their weighted versions), and · is the inner product in Rd (see [30, 39,

40, 18]). Using the process r(t, u), the price at time t of a zero-coupon bond (see [31])with maturity T is

BT (t) = e−∫ T−t0

r(t,u)du.

The model can be used to price swaptions, caps, and other interest rates and currencyderivatives.

The solution rt of (1) is a Markov process. This makes a connection with stochasticoptimal control and partial differential equations. Let us describe it first in a simple caseof European options. Given a contingent claim with the pay-off function F : H → R andan initial curve at time t, rt(u) = x(u), the rational price of the option maturing at timeT is

V (t, x) = E

(e−

∫ Tt

r(s,0)dsF (rT ) : rt(u) = x(u)). (2)

For instance for a European swaption on a swap with cash-flows Ci, i = 1, ..., n at timesT < T1 < ... < Tn,

F (z) =

(K −

n∑i=1

Cie∫ Ti−T0 z(u)du

)+

(3)

for some K > 0. The function V given by the Feynman-Kac formula (2) should satisfythe partial differential equation

∂V

∂t+

1

2

d∑i=1

< D2V τi(t), τi(t) > + < c(τt)− Ax,DV > −x(0)V = 0

V (T, x) = F (x), (t, x) ∈ (0, T )×H,

(4)

where < ·, · > is the inner product in H . The above equation is an infinite-dimensionalBlack-Scholes equation. It was analyzed in [17] in the space H = L2((0,+∞)) whereexistence of smooth solutions was proved for smooth F and τ independent of t but possiblydepending on the state variable x. The existence of solutions was also shown for somenonsmooth F when τ was a constant by an argument that allowed a parallel between (4)and a finite dimensional Black-Scholes equation (see also [13, 40]).

2

Many other problems of mathematical finance can be recast in the framework of theMusiela model as stochastic optimal control problems. The dynamic programming anal-ysis of them gives rise to various infinite-dimensional partial differential equations. Forinstance the problem of pricing of American options can be rephrased as an optimalstopping problem for an infinite-dimensional diffusion process and is connected with anobstacle problem

max

∂V

∂t+

1

2

d∑i=1

< D2V τi(t), τi(t) > + < c(τt)− Ax,DV > −x(0)V, V − ϕ

= 0 (5)

for some function ϕ. This equation was studied in [41] from the point of view of Bellman’sinclusions and a similar obstacle problem for a related model was also investigated in [16].

One of the drawbacks of the Musiela model is that it does not guarantee the positivityof rates and in some cases it is almost certain that they are not positive (see [40]). Toavoid such possibilities the term x(0) was replaced by x+(0) in [41]. We do the same hereand throughout the paper we always take the positive part of the rates. This leads toHJB equations with the right negative coefficient in front of the zero order terms.

A very interesting problem is the question of robustness of the model. Suppose thatthe dynamics of the forward rates are given by equation (1), however we are not able todetermine precisely the process τ that describes the volatility of the market. We onlyknow that it takes values in some set Σ ⊂ Hd. We consider an agent who wants to priceand hedge a European contingent claim with pay-off F (rT ) that depends on the valueof the forward rate curve at the maturity time T . An approach relying on earlier works[1, 29, 14, 36] was proposed in [40] and a similar strategy in the finite dimensional casewas also studied in [23, 24]. It is based on building a so called superhedging strategy andit fixes the price Ct of the option at time t as Ct = V (t, rt), where V is the solution of theHJB equation

∂V

∂t− < Ax,DV > +H(DV,D2V )− x+(0)V = 0 in (0, T )×H

V (T, x) = F (x) in H,

(6)

where for p ∈ H and X ∈ S(H), the space of bounded self-adjoint operators in H ,

H(p,X) = supτ∈Σ

1

2

d∑i=1

< X τi, τi > + < c(τ), DV >

.

Equation (6) is an infinite dimensional Black-Scholes-Barenblatt (BSB) equation associ-ated to the contingent claim F . An interesting thing is that it has a stochastic optimalcontrol interpretation as the dynamic programming equation for the problem in which(given an initial condition rt ∈ H) we try to maximize the pay-off

E

(e−

∫ Tt r+(s,0)dsF (rT )

)(7)

3

with respect to all progressively measurable stochastic processes τ taking values in Σ. Theprocesses τ become controls and the maximization of (7) should give the value function V .We do not deal here with the robustness problem and it seems that the construction of thesuperhedging strategy can be carried out only in special cases. However the investigationof the BSB equation (6) is of fundamental importance. We remind that we have takenr+ in (7) (and consequently x+ in (6)) to eliminate the possibility of negative rates.

Equation (6) has not been investigated in the literature. Moreover in cases of interestin finance the pay-off function F is not even C1 and so a notion of a generalized solutionseems to be needed. In this paper we study this equation from the viscosity solutionspoint of view. We follow the B-continuous solution approach of [9, 10, 38]. We prove thatequation (6) has a unique uniformly continuous viscosity solution that coincides with thevalue function provided by the maximization of (7). The existence of solutions is shownin two cases; when F is B-continuous, and when F is weakly continuous. In the first casewe use directly the dynamic programming principle, and in the second case we employ aGalerkin-type approximation similar to those in [9, 38]. We investigate equation (6) inthe space H = H1(R+) which seems to provide the right set-up because of the term x+(0)in the equation. One can also investigate it in veighted versions of H1(R+).

We refer the reader to [2, 11, 3, 4, 5, 6, 19, 20] for results on regular solutions ofunbounded second order HJB equations in infinite dimensions and to [21, 22, 25, 27, 32, 33]for other results for such equations using viscosity solutions. The reader can also consult[7, 15, 42] for the overview of the theory of viscosity solutions in finite dimensional spacesand its connection with stochastic optimal control.

2 Preliminaries

Throughout this paper H will be the Sobolev space H1([0,+∞)). We will denote by< ·, · > and ‖ · ‖ respectively the inner product and the norm in H . Given a Banachspace V we denote by C1,2((0, T ) × V ) the set of all functions defined on (0, T ) × V,differentiable in the first variable, twice Frechet differentiable in the second variable, andhaving continuous derivatives, and we denote by L(V ) the space of all bounded linearoperators in V . We denote by BR the closed ball of radius R centered at the origin in H .We recall that a modulus is a nondecreasing, subadditive function

ω : R+ → R

+

such that ω(0+) = 0. As a consequence of the subadditivity, given ε > 0 there is a positiveconstant Kε, such that

ω(r) ≤ ε+Kεr ∀r ≥ 0.

A local modulus is a functionσ : R

+ ×R+ → R

+,

non-decreasing in both variables, sub-additive in the first variable, and such that σ(0+, R) =0 for every R > 0.

4

Denote

V =

x ∈ H :

√ux(u),

√udx

du(u) ∈ L2((0,∞))

.

The space V will be equipped with the norm

‖x‖2V =

∫ ∞

0

((1 + u)(x2(u) + (

dx

du(u))2

)du.

We denote by Σ a fixed bounded and closed subset of Vd. This will be a standingassumption that will not be repeated throughout the paper. We will make assumptionsabout Σ only when we require something more.

A stochastic base is any five-tuple ν = (Ων , (Fνt ),Pν ,W ν

t , τν), where (Ων , (Fν

t ),Pν)is a filtered probability space, (W ν

t ) is an Rd-valued Brownian motion, the filtration

(Fνt ) is generated by the Brownian motion, and τ ν is a progressively measurable process

(with respect to the given filtration) which takes values in Σ. The set of all (admissible)stochastic bases will be denoted by Θ.

The following lemma is needed to ensure that the stochastic differential equation(SDE) below and the control problem associated with the BSB equation are well posedin H .

Lemma 2.1. The function c : Vd → H defined by

c(x)(u) =d∑

k=1

xk(u)

∫ u

0

xk(µ)dµ

is locally Lipschitz.

Proof of lemma. Let R > 0 and x, y ∈ Vd be such that ‖xk‖V, ‖yk‖V ≤ R, fork = 1, ..., d. Then

|c(x)(u)− c(y)(u)|2 ≤d∑

k=1

2d

(u|xk(u)− yk(u)|2

∫ u

0

x2k(µ)dµ

+ uy2k(u)

∫ u

0

|x(µ)− y(µ)|2dµ).

Integrating we have∫ +∞

0

|c(x)(u)− c(y)(u)|2du ≤d∑

k=1

2dR2

∫ +∞

0

(1 + u)|(xk − yk)(u)|2du.

Similarly we obtain

|(c(x))′(u)− (c(y))′(u)|2 ≤ 3d

d∑k=1

(4R2|xk(u)− yk(u)|2

+ u((yk)′)2(u)

∫ u

0

|x(µ)− y(µ)|2dµ

+ u|(xk)′(u)− (yk)

′(u)|2

∫ u

0

x2k(µ)dµ

)

5

which after integration yields

∫ +∞

0

|(c(x))′(u)− (c(y))′(u)|2du ≤3d(4R2 + 1)

d∑k=1

(∫ +∞

0

|(xk − yk)(u)|2du

+

∫ +∞

0

u|(xk)′(u)− (yk)

′(u)|2du

).

The claim now follows easily.

Therefore, given a stochastic base ν the stochastic partial differential equationdrs = (−Ars + c(τs))ds+ τs · dWs, s ∈ (t, T ]rt = x

(SDE)

has a unique solution (see [12]).Our control problem consists in maximizing the cost functional (7) over all admissible

stochastic bases. This defines a value function

V (t, x) = supν∈Θ

Eν(F (rT )e−

∫ Tt r+

u (0)du).

By Bellman’s optimality principle the value function V should solve the Hamilton-Jacobi-Bellman equation (the BSB equation)

∂V

∂t− < Ax,DV > +H(DV,D2V )− x+(0)V = 0 in (0, T )×H

V (T, x) = F (x) in H,

(CP )

where for p ∈ H and X ∈ S(H)

H(p,X) = supτ∈Σ

1

2

d∑i=1

< X τi, τi > + < c(τ), DV >

.

We will make this precise in the sequel.The operator A = − d

duis a maximal monotone operator in H . By [35] there exists a

bounded, self-adjoint, positive operator B such that

A∗B is bounded and A∗B +B ≥ 0. (8)

The operator B can be obtained from the polar decomposition of A. For α > 0 we denoteby H−α the completion of H under the norm ||x||2−α = 〈Bαx, x〉, and by Hα we denotethe space B

α2 (H) equipped with the norm ||x||2α = 〈B−αx, x〉. We remind that Hα is the

dual of H−α.

Definition 2.1. A continuous function u : (0, T )× H → R is said to be B-upper semi-continuous if u(t, x) ≥ lim supn→∞ u(tn, xn) whenever xn x, tn → t, Bxn → Bx.

6

A continuous function v : (0, T ) × H → R is said to be B-lower semi-continuous ifv(t, x) ≤ lim infn→∞ v(tn, xn) whenever xn x, tn → t, Bxn → Bx.A B-continuous function is a function which is both B-upper semi-continuous and B-lowersemi-continuous.

Remark 2.1. It is clear that a weakly continuous function is B-continuous and also B-continuity is the same as uniform continuity on bounded and closed subsets of (0, T )×Hin the topology inherited from (0, T )×H−1.

We now recall the definition of viscosity solution of (CP ) (see [10, 38]). We need twotypes of test functions.

Type 1 : Functions ϕ : (0, T ) × H → R which are in C1,2((0, T ) × H) and such thatϕ is B-continuous, and ∂ϕ

∂t, A∗Dϕ, Dϕ, D2ϕ are uniformly continuous on closed subsets

of (0, T )×H . Note that this requires that Dϕ takes values in the domain of A∗. Type 2 :

Functions g : H → R which are radial, nondecreasing, twice Frechet differentiable, andsuch that Dg, D2g are uniformly continuous on H .

Definition 2.2. A B-upper semi-continuous function u is a viscosity subsolution of (CP )if

∂ϕ

∂t(t, x)−〈x,A∗Dϕ(t, x)〉 − x+(0)u(t, x)

+H(Dϕ(t, x) +Dg(x), D2ϕ(t, x) +D2g(x)) ≥ 0

whenever u− (ϕ+ g) has a global maximum at the point (t, x).A B-lower semi-continuous function v is a viscosity supersolution of (CP ) if

∂ϕ

∂t(t, x)−〈x,A∗Dϕ(t, x)〉 − x+(0)v(t, x)

+H(Dϕ(t, x)−Dg(x), D2ϕ(t, x)−D2g(x)) ≤ 0

whenever v − (ϕ− g) has a global minimum at the point (t, x).A viscosity solution of (CP ) is a function which is both a viscosity subsolution and aviscosity supersolution. In the above definition ϕ is a test function of Type 1 and g is atest function of Type 2.

Remark 2.2. It follows from the above definition that a smooth solution of (CP ) in thesense of the definition of [17] is also a viscosity solution of (CP ). Also we could replacein the definition of the test functions “uniformly continuous” by “uniformly continuouson bounded sets and having polynomial growth” without altering the results of the paper.Moreover the maxima and minima in the definition can be assumed to be strict (see [38]).

Definef(x) = x+(0) (9)

7

for x in H . Notice that f is a weakly continuous function on H and so it is B-continuous.Moreover it is easy to see that f has at most linear growth at infinity and for instance

x+(0) ≤ 2‖x‖. (10)

We will be assuming that either

(A) : F is bounded and B-continuous.

or(B) : F is bounded and weakly continuous.

Note that (A) is weaker than (B). Assumption (B) holds in many interesting cases, forexample if F is given by (3).

3 Comparison

In this section we prove a comparison result for bounded viscosity solutions of (CP ).

Theorem 3.1. Let (A) hold and let Σ be a compact subset of Hd−1. Let u be a viscosity

subsolution of (CP ), and v be a viscosity supersolution of (CP ) such that

u,−v ≤M for some M > 0 (11)

and

limt→T

(u(t, x)− F (x))+ + (v(t, x)− F (x))− = 0,

uniformly on bounded sets. (12)

Then for every 0 < T1 < T

limR↑∞

lim(r,η)↓(0,0)

u(t, x)− v(s, y) : |t− s| < η, ‖x− y‖−1 < r

x, y ∈ BR, T1 ≤ t, s ≤ T ≤ 0. (13)

In particular u ≤ v.

Proof. Let 0 < T1 < T . We argue by contradiction. Assume that (13) is not true.Then there exists γ > 0 such that

2γ < limR↑∞

lim(r,η)↓(0,0)

u(t, x)− v(s, y) : |t− s| < η, ‖x− y‖−1 < r

x, y ∈ BR, T1 ≤ t, s ≤ T.It is then possible to find σ > 0 such that

γ < limR↑∞

lim(r,η)↓(0,0)

uσ(t, x)− vσ(s, y) : |t− s| < η, ‖x− y‖−1 < r,

x, y ∈ BR, (14)

8

where uσ(t, x) = u(t, x)− σt

and vσ(s, y) = v(s, y) + σs.

Let us select a radial function µ : H → R, nondecreasing, linearly growing, andsuch that Dµ,D2µ are bounded and uniformly continuous. (We could take for exampleµ(x) =

√1 + 〈x, x〉.) Let

m := limR↑∞

lim(r,η)↓(0,0)

supuσ(t, x)− vσ(s, y) : |t− s| < η, ‖x− y‖−1 < r

x, y ∈ BR, (15)

mα := lim(r,η)↓(0,0)

supuσ(t, x)− vσ(s, y)− αµ(x)− αµ(y) : |t− s| < η, ‖x− y‖−1 < r,

mα,ε,β := supuσ(t, x)− vσ(s, y)− αµ(x)− αµ(y)− 1

2ε‖x− y‖2

−1 −1

2β|t− s|2,

mα,ε,β,ζ := inf|a|+|b|+‖p‖+‖q‖<ζ

supuσ(t, x)− vσ(s, y)− αµ(x)− αµ(y)− 1

2ε‖x− y‖2

−1

− 1

2β|t− s|2 + at+ bs+ 〈Bp, x〉+ 〈Bq, y〉.

We have (see [25])

m = limα↓0

mα, (16)

mα = lim(ε,β)↓(0,0)

mα,ε,β, (17)

mα,ε,β = limζ↓0

mα,ε,β,ζ. (18)

Using a perturbed optimization result as in [10] (based on results of [37] and the factthat the function involved is B-upper semi-continuous) there exist sequences an, bn ∈ R,and pn, qn ∈ H such that |an|+ |bn|+ ‖pn‖+ ‖qn‖ → 0 as n→∞, and such that

uσ(t, x)− vσ(s, y)− αµ(x)− αµ(y)− 1

2ε‖x− y‖2

−1 −1

2β|t− s|2 + ant+ bns

+ 〈Bpn, x〉+ 〈Bqn, y〉achieves a strict global maximum at some point (t, s, x, y) ∈ (0, T ] × (0, T ] × H × H.Convergencies (16)-(18) yield

limβ↓0

lim supn→∞

1

2β|t− s|2 = 0, (19)

limε↓0

limβ↓0

lim supn→∞

1

2ε‖x− y‖2

−1 = 0, (20)

limα↓0

lim(ε,β)↓(0,0)

lim supn→∞

(αµ(x) + αµ(y)) = 0. (21)

9

By the definition of uσ and vσ we have t > 0 and s > 0. In light of (12), (19), (20), andthe B-continuity of F it also follows that t < T and s < T for big n and small β, ε.

Choose a basis e1, e2, . . . , en, . . . of H−1 made of elements of H . Given N ≥ letHN = spane1, e2, . . . , eN, and let PN denote the orthogonal projection of H−1 onto HN .Denote QN = I − PN . We have

‖x− y‖2−1 = ‖PN(x− y)‖2

−1 + ‖QN(x− y)‖2−1

and

‖QN (x− y)‖2−1 ≤ 2〈BQN(x− y), x− y〉+ 2‖QN(x− x)‖2

−1

+ 2‖QN(y − y)‖2−1 − ‖QN(x− y)‖2

−1

with equality at x, y. Therefore defining

u1(t, x) = uσ(t, x)− 〈BQN(x− y), x〉ε

− ‖QN(x− x)‖2−1

ε

+‖QN(x− y)‖2

−1

2ε− αµ(x) + ant+ 〈Bpn, x〉

and

v1(s, y) = vσ(s, y)− 〈BQN (x− y), y〉ε

+‖QN(y − y)‖2

−1

ε+ αµ(y)− bns− 〈Bqn, y〉.

we have that

u1(t, x)− v1(s, y)− 1

2ε‖PN(x− y)‖2

−1 −1

2β|t− s|2

has a strict global maximum at (t, s, x, y). At this step we need to produce appropriatetests functions to be able to use the definition of solution. This is done using partialsup-convolution techniques (see [8], [28], [38]).

Lemma 3.1. Given N ≥ 1 there exist functions ϕk, ψk,∈ C1,2((0, T ) × H−1) with uni-formly continuous derivatives such that

u1(t, x)− ϕk(t, x)

has a global maximum at some point (tk, xk),

v1(s, y) + ψk(s, y)

has a global minimum at some point (sk, yk), and

10

(tk, xk, u1(tk, xk),

∂ϕk

∂t(tk, xk), Dϕk(tk, xk), D

2ϕk(tk, xk))→

(t, x, u1(t, x),

t− s

β,BPN (x− y)

ε,XN

), (22)

(sk, yk, v1(sk, yk),

∂ψk

∂t(sk, yk), Dψk(sk, yk), D

2ψk(sk, yk))→

(s, y, v1(s, y)

s− t

β,BPN(y − x)

ε, YN

), (23)(

XN 00 YN

)≤ 2

ε

(B −B−B B

), (24)

with the convergencies being in R×H × R×R×H2 × L(H).

Proof of lemma. Denote xN = PNx, x⊥N = QNx, yN = PNy, y

⊥N = QNy and define

u1(t, xN) = supx⊥N∈QNH

u1(t, xN + x⊥N),

v1(s, yN) = infy⊥N∈QN H

v1(s, yN + y⊥N),

the partial sup- and inf-convolutions of u1 and v1 respectively. Then

(u1)∗(t, xN)− (v1)∗(s, yN)− 1

2ε‖xN − yN‖2

−1 −1

2β|t− s|2 (25)

has a strict global maximum over (0, T )× (0, T )×HN ×HN at (t, s, xN , yN), where (u1)∗

and (v1)∗ are the upper and lower semi-continuous envelopes of u1 and v1. Moreover wehave (u1)

∗(t, xN) = u1(t, x), (v1)∗(s, yN) = v1(s, y).We can now apply the finite dimensional maximum principle (see [7]) to (25) when we

consider (0, T )×HN as a space with the topology inherited from (0, T )×H−1 (which isequivalent to the topology inherited from (0, T )×H). Denote HN with this topology byHN . Therefore there exist bounded functions ϕk, ψk,∈ C1,2((0, T )× HN) with uniformlycontinuous derivatives such that (u1)

∗(t, xN)− ϕk(t, xN) has a strict global maximum atsome point (tk, xk

N), (v1)∗(s, yN) + ψk(s, yN) has a strict global minimum at some point(sk, yk

N), and such that as k →∞(tk, xk

N , (u1)∗(tk, xk

N),∂ϕk

∂t(tk, xk

N ), DHNϕk(t

k, xkN ), D2

HNϕk(t

k, xkN))→

(t, xN , u1(t, xN),

t− s

β,xN − yN

ε, XN

),

(sk, yk

N , (v1)∗(sk, ykN),

∂ψk

∂t(sk, yk

N), DHNψk(s

k, ykN), D2

HNψk(s

k, ykN))→

(s, yN , v1(s, yN),

s− t

β,yN − xN

ε, YN

),(

XN 0

0 YN

)≤ 2

ε

(I −I−I I

)in H−1 ×H−1

11

for some N×N matrices XN and YN that as operators in L(H−1) satisfy XN = PNXNPN ,YN = PNYNPN and are symmetric. (Above the symbols DHN

andD2HN

denote the Frechet

derivatives in HN .) Since in HN the topology of H−1 is equivalent to the topology of Hthe above convergencies hold in the topology of R × H × R × R × H × L(H). We nowextend ϕk, ψk to functions in C1,2((0, T )×H−1) by setting ϕk(t, x) = ϕk(t, PNx), ψk(s, y) =ψk(s, PNy). Then, if DH−1ϕk denotes the Frechet derivative of ϕk in H−1 we have

Dϕk(t, x) = BDH−1ϕk(t, x), D2ϕk(t, x) = BD2

H−1ϕk(t, x) (26)

and the same also holds for ψk.We now put everything back into H . For k fixed we drop the upper and lower en-

velopes, use a perturbed optimization result result as before to find sequences aj, bj ∈ R,pj , qj ∈ H , such that |aj|+ |bj |+ ‖pj‖+ ‖qj‖ → 0 an j →∞, and

u1(t, x)− ϕk(t, x)− ajt− 〈Bpj , x〉has a global maximum at some point (tj, xj), (27)

v1(s, y) + ψk(s, PNy)− bjs− 〈Bqj , y〉has a global minimum at some point (sj , yj). (28)

Combining (27) and the fact that (u1)∗(t, xN) − ϕk(t, xN ) has a strict global maximum

at (tk, xkN ), we can deduce that (tj , PNxj) → (tk, xk

N) and u1(tj, PNxj) → (u1)∗(tk, xk

N)as j → +∞. Similarly (28) and the fact that (v1)∗(s, yN) + ψk(s, yN) has a strict globalminimum at some point (sk, yk

N) implies that (sj, PNyj) → (sk, ykN) and v1(sj, PNyj) →

(v1)∗(sk, yk

N) as j → +∞. It then easily follows by selecting an appropriate subsequencejk that xjk

→ x and yjk→ y (see [8] for details). Convergencies (22), (23), and the

inequality (24) now follow from this, (26), and the convergencies for ϕk and ψk.

We now return to the proof of the theorem. Since u is a subsolution and v is asupersolution of (CP ), using (27), (28), (22), (23), and letting k →∞ we get

an − t− s

β+σ

t2+ f(x)u(t, x) + 〈x, A

∗B(x− y)

ε〉 − 〈x, A∗Bpn〉

− H(αDµ(x)−Bpn +

B(x− y)

ε, αD2µ(x) +XN +

BQN

ε

)≤ 0

(29)

and

−bn − t− s

β− σ

s2+ f(y)v(s, y) + 〈y, A

∗B(x− y)

ε〉+ 〈y, A∗Bqn〉

− H(−αDµ(y) +Bqn +

B(x− y)

ε,−αD2µ(y)− YN − BQN

ε

)≥ 0.

(30)

Note that because of our choice of the cut-off function µ, x and y remain in a bounded

12

ball BRα , where Rα depends exclusively on α. Note also that

|H(p,X)−H(q, Y )| ≤ C(‖p− q‖+ ‖X − Y ‖),

where C depend on the bound on Σ in V. Combining those facts with (29) and (30)yields

2σ

T 2+ f(x)u(t, x)− f(y)v(s, y) + 〈x− y,

A∗B(x− y)

ε〉 ≤ 2CαL

+ωα(n)−H(B(x− y)

ε,XN +

BQN

ε

)+H

(B(x− y)

ε,−YN − BQN

ε

),

(31)

where L = supx∈H‖Dµ(x)‖∞+‖D2µ(x)‖∞ and ωα(n) → 0 as n→∞. We observe that

−H(p,X) +H(p, Y ) ≤ supτ∈Σ

d∑i=1

〈(Y −X)τi, τi〉. (32)

The function f is weakly continuous and therefore it is B-continuous. Combining thiswith (11), the fact that f is nonnegative and that u(t, x) > v(s, y) we obtain

f(x)u(t, x)− f(y)v(s, y) = (v(s, y)(f(x)− f(y)) + f(x)(u(t, x)− v(s, y))

≥ −Mωf (‖x− y‖−1, Rα),(33)

where ωf is the local modulus of continuity of f in the ‖ · ‖−1 norm . The compactness ofΣ in Hd

−1 implies that

supτ∈Σ

d∑i=1

‖QNτi‖2−1 → 0 as N →∞. (34)

Inequality (24) implies that XN + YN ≤ 0. This, together with (31)-(34), gives

2σ

T 2−Mωf(‖x− y‖−1, Rα)− ‖x− y‖2

−1

ε≤

ωα(n) + 2CαL+2

εsupτ∈Σ

d∑i=1

‖QNτi‖2−1.

We now take limN→∞, lim supn→∞, limβ↓0, limε↓0, limα↓0 in that order to arrive at σ ≤ 0which is a contradiction.

In the next section we will establish some properties of the value function and we willuse the dynamic programming principle to prove that the value function is a viscositysolution of (CP ).

13

4 Existence of solutions: F is B-continuous

Throughout this section we assume that (A) holds. We will start recalling some propertiesof the solution of (SDE). We remind that given a stochastic base ν, there exists a uniquemild-solution of the stochastic differential equation (SDE) that is given by

rs = e−(s−t)Ax+

∫ s

t

e−(s−u)Ac(τu)du+

∫ s

t

e−(s−u)Aτu · dWu, (35)

where e−tA is the semi-group of contractions generated by −A. Let An = A(I + An)−1 be

the Yosida approximation of A, and let rns be the unique solution of

drns = (−Anr

ns + c(τs))ds+ τs · dWs, s ∈ (t, T ]

rnt = x.

(SDEn)

It is known (see [12]) or it can be deduced from the solution formula (35) that

limn→∞

E(

supt≤s≤T

‖rns − rs‖2

)= 0, (36)

supt≤s≤T

E‖rs‖2p ≤ CT,p(1 + ‖x‖2p) for p ≥ 1, (37)

E‖rs − x‖2 ≤ CT (‖e−(s−t)Ax− x‖2 + (s− t)), (38)

where CT,p and CT are constants depending exclusively on T, p, and the bound on τ .Moreover the constants in (36), (37), (38) are independent of the choice of the stochasticbase. This fact will be used in the proofs in this section.

We will need the following lemmas:

Lemma 4.1. Let Xs and Ys be solutions to (SDE) with respective initial data Xt =x, Yt = y. Then

supt≤s≤T

E‖Ys −Xs‖2−1 ≤ CT‖x− y‖2

−1, (39)

E‖Xs − x‖2−1 ≤ Cr,T (s− t) if ‖x‖ ≤ r, (40)

where CT is a constant depending on T, and Cr,T is a constant depending on r, T . Bothconstants are independent of the choice of the stochastic base.

Proof. Let XnS and Y n

S be the corresponding solutions of (SDEn). Let Zns = Xn

s −Y ns .

Applying Ito’s formula to the function ‖z‖2−1, we have

E‖Zns ‖2

−1 = ‖x− y‖2−1 + 2E

∫ s

t

〈Znu , A

∗nBZ

nu 〉du (41)

Let Zs = Xs − Ys. For fixed u ∈ [t, T ], using (36) and the fact A∗nx converges to A∗x for

every x ∈ D(A∗), we conclude that

limn→+∞

〈Znu , A

∗nBZ

nu 〉 = 〈Zu, A

∗BZu〉 P− a.s.

14

Moreover, using (36) and (37), we obtain

E|〈Znu , A

∗nBZ

nu 〉|2 ≤ CT (1 + ‖x‖2 + ‖y‖2)2

for some constant CT independent of u. Therefore by the Lebesgue dominated convergencetheorem

limn→+∞

E〈Znu , A

∗nBZ

nu 〉 = E〈Zu, A

∗BZu〉.

Similarly we have limn→∞

E‖Zns ‖2

−1 = E‖Zs‖2−1. Thus, letting n→∞ in (41) and using (8),

it follows that

E‖Xs − Ys‖2−1 ≤ ‖x− y‖2

−1 +

∫ s

t

E‖Xu − Yu‖2−1du,

and then Gronwall’s inequality gives (39).Similarly,

Xns − x =

∫ s

t

(−AnXnu + c(τu)

)du+

∫ s

t

τu · dWs.

Using Ito’s formula we have

E‖Xns − x‖2

−1 = 2

∫ s

t

E

(〈−Xn

u , A∗nB(Xn

u − x)〉 + 〈c(τu), B(Xnu − x)〉

+1

2

d∑i=1

‖τi(u)‖2−1

)du.

We can then use (36), (37), let n→ +∞, and argue as before to get

E‖Xs − x‖2−1 = 2

∫ s

t

E

(〈−Xu, A

∗B(Xu − x)〉+ 〈c(τu), B(Xu − x)〉

+1

2

d∑i=1

‖τi(u)‖2−1

)du.

Note that in light of (37) the integrand on the right-hand side of the previous inequalityremains bounded by some constant Cr,T when x remains in Br. Inequality (40) now fol-lows easily.

The next lemma establishes Ito-type formulas for our test functions.

Lemma 4.2. Let rs be a solution of (SDE). If ϕ is a test function of Type 1 then

E(e−

∫ st

f(ru)duϕ(s, rs))

= E

∫ s

t

e−∫ u

tf(rµ)dµ

(∂ϕ∂t

(u, ru)− 〈ru, A∗Dϕ(u, ru)〉

− f(ru)ϕ(u, ru) +Hτu(Dϕ(u, ru), D2ϕ(u, ru))

)du

+ ϕ(t, x).

15

If g is a test function of Type 2 then

E(e−

∫ st f(ru)dug(rs)

) ≤ E

∫ s

t

e−∫ ut f(rµ)dµ

(− f(ru)g(ru)

+Hτu(Dg(ru), D2g(ru))

)du+ g(x),

where Hτ (p,X) = 〈c(τ), p〉 + 12

∑di=1〈Xτi, τi〉 for every p ∈ H, τ ∈ V, and for every

bounded self-adjoint operator X on H.

Proof. Let rns be the solution of (SDEn), and let ϕ be a function of Type 1. Using

Ito’s formula and Ito’s product rule we have

E(e−

∫ st f(rn

u )duϕ(s, rns ))

= E

∫ s

t

e−∫ u

t f(rnµ)dµ(∂ϕ∂t

(u, rnu)− 〈rn

u , A∗nDϕ(u, rn

u)〉− f(rn

u)ϕ(u, rnu) +Hτu(Dϕ(u, rn

u), D2ϕ(u, rnu)))du

+ ϕ(t, x).

Note that all functions that appear in the above integrand are of polynomial growth.Therefore, using (36), (37), and arguing as in the proof of Lemma 4.1 we can let n→ +∞to reach the desired conclusion. The proof for the test function g is similar. The inequalityis a consequence of the monotonicity of both g and A∗. As before we have

E(e−

∫ st f(rn

u )dug(rns ))

= E

∫ s

t

e−∫ ut f(rn

µ)dµ(− f(rn

u)g(rnu)− 〈rn

u , A∗nDg(r

nu)〉

+Hτu(Dg(rnu), D2g(rn

u)))du+ g(x).

Since g is of the form g(x) = g0(‖x‖) for some nondecreasing function g0 in C2([0,+∞)),we get

−〈x,A∗nDg(x)〉 = −g

′0(‖x‖)‖x‖ 〈x,A∗

nx〉 ≤ 0.

We then deduce that

E(e−

∫ st

f(rnu )dug(rn

s )) ≤ E

∫ s

t

e−∫ u

tf(rn

µ)dµ(− f(rn

u)g(rnu)

+Hτu(Dg(rnu), D2g(rn

u)))du+ g(x)

and let n→ +∞ to obtain the result.

The next proposition establishes the continuity properties of the value function.

Proposition 4.1. Assume that (A) holds. Let K be a positive real number such that

|F (x)| ≤ K for every x ∈ H.Then there exists a local modulus σ such that

|V (t, x)− V (s, y)| ≤ σ(|t− s|+ ‖x− y‖−1;R

)for all 0 ≤ t, s ≤ T, x, y ∈ BR.

16

Proof. We first prove the continuity in the state variable x. Let Xs and Ys be solutionsof (SDE) with respective initial data Xt = x, Yt = y, x, y ∈ BR. Then

|V (t, x)− V (t, y)| ≤supν

Eν |F (XT )− F (YT )|+

Ksupν

∫ T

t

Eν |f(Xu)− f(Yu)|du.

(42)

Given ε > 0,

Eν(|F (XT )− F (YT )|) ≤ E

ν(|F (XT )− F (YT )|IXT ,YT ,∈B 1

ε)

+K(P ν(‖XT‖ > 1

ε) + P ν(‖XT‖ > 1

ε))

≤ Eν(ωF (‖XT − YT‖−1,

1

ε))

+ εCT (1 +R)

≤ KεEν(‖XT − YT‖−1) + εCT,R

≤ Cε‖x− y‖−1 + εCT,R,

(43)

where ωF denotes the local modulus of continuity of F in the topology ofH−1 (see Remark

2.1). A similar upper bound can be obtained for∫ T

tE

ν |f(Xu) − f(Yu)|du. Since theseestimates are independent of the stochastic base ν we deduce that

|V (t, x)− V (t, y)| ≤ Cε‖x− y‖−1 + εCT,R.

Defining ω(r, R) = infε>0Cεr + εCT,R we conclude that

|V (t, x)− V (t, y)| ≤ ω(‖x− y‖−1, R)

which concludes the first part of the proof.To prove the continuity in t let us fix x ∈ BR and let s > t. Let Xu be the solution of

(SDE) with initial data Xt = x, and let Xu be the solution of (SDE) with initial dataXs = x. Then

|V (t, x)− V (s, x)| ≤ supν

Eν∣∣e− ∫ T

tf(Xu)duF (XT )− e−

∫ Ts

f(Xu)duF (XT )∣∣

≤ Ksupν

Eν∣∣e− ∫ s

tf(Xu)du − 1

∣∣+Ksup

ν

∫ T

s

Eν |f(Xu)− f(Xu)|du+ sup

νE

ν |F (XT )− F (XT )|.

(44)

Using (10) and (37) we obtain

Eν∣∣e− ∫ s

tf(Xu)du − 1

∣∣ ≤ Eν∣∣ ∫ s

t

f(Xu)du∣∣ ≤ 2

∫ s

t

Eν‖Xu‖du ≤ CR(s− t).

Using the argument of (43) we get

Eν |F (XT )− F (XT )| ≤ CεE

ν‖Xs − x‖−1 + CRε

17

Therefore, by (40), it follows that

supν

Eν |F (XT )− F (XT )| ≤ Cε|s− t| 12 + CRε

for every ε > 0. Finally, using (10) and the B-continuity of f , and arguing similarly as

above we can obtain the same estimate for supν

∫ T

sE

ν |f(Xu)− f(Xu)|du. The proof nowfollows.

In the next theorem we prove that the value function is the unique viscosity solution of(CP ). We will use the dynamic programming principle that we state below in its simplestform,

V (t, x) = supν∈Θ

(e∫ t+ht

f(rs)dsV (t+ h, rt+h))

∀h > 0 (DPP ).

We do not prove it here. Since we are using relaxed controls and the value function iscontinuous the proof follows the one presented in [42].

Theorem 4.1. Let (A) hold and let Σ be a compact subset of Hd−1. Then V is the unique

viscosity solution of (CP ) among all bounded solutions that satisfy (12).

Proof. The uniqueness follows from Theorem 3.1. We will only prove that V is aviscosity subsolution since the proof that V is a supersolution is very similar and simpler.

Let ϕ be a test function of Type 1, let g be a test function of Type 2, and let V (s, y)−ϕ(s, y) − g(y) have a global maximum at (t, x). Using (DPP ), for every h > 0, thereexists a stochastic base νh such that

V (t, x)− h2 ≤Eνh(e−

∫ t+ht f(rh

u)duV (t+ h, rht+h)

)≤E

νh(e−

∫ t+ht

f(rhu)du(ϕ(t+ h, rh

t+h) + g(rht+h))

),

where rhs is the solution of (SDE) given the stochastic base νh. Using Lemma 4.2 we then

have

−h ≤ 1

h

∫ t+h

t

Eνh

(∂ϕ

∂t(u, rh

u)− 〈A∗Dϕ(u, rhu), r

hu〉

+Hτhu

(Dϕ(u, rh

u) +Dg(rhu), D2ϕ(u, rh

u) +D2g(rhu))

− f(rhu)(ϕ(u, rh

u) + g(rhu)))e−

∫ ut f(rh

µ)dµdu.

(45)

The strategy is to replace first e−∫ u

t f(rhµ)dµ by 1 in (45). Let us denote

Mh(u) =∂ϕ

∂t(u, rh

u)− 〈A∗Dϕ(u, rhu), r

hu〉

+Hτhu

(Dϕ(u, rh

u) +Dg(rhu), D2ϕ(u, rh

u) +D2g(rhu))

− f(rhu)(ϕ(u, rh

u) + g(rhu)).

18

By the properties of ϕ and g, there exists a constant C (depending only on ϕ and g) anda positive integer m such that

|Mh(u)| ≤ C(1 + um + ‖rhu‖m).

Therefore, using (37) and (10), we obtain

Eνh∣∣Mh(u)

(e−

∫ ut f(rh

µ)dµdu− 1)∣∣ ≤ Cm,T (1 + Tm + ‖x‖m)(u− t).

In light of the above inequality (45) can be rewritten as

−h ≤ 1

h

∫ t+h

t

Eνh

Mh(u)du+ σ(h), (46)

where σ(h) → 0 as h → 0. The next step is to replace u by t and rhu by x in (46) and

estimate the error we make while doing so. We will only show how to estimate one termas all estimates are similar. Denote by σ the modulus of continuity of A∗Dϕ. Then using(37) and (38) we have

1

h

∣∣ ∫ t+h

t

Eνh(〈A∗Dϕ(u, rh

u), rhu〉 − 〈A∗Dϕ(t, x), x〉)du∣∣

≤ 1

h

∫ t+h

t

C(E

νh‖rhu − x‖2

) 12 + ‖x‖Eνh(

σ(|u− t|+ ‖rhu − x‖))du

≤ ω(h),

where limh→0 ω(h) = 0. The other terms in (46) can be treated in a similar way to yield

−h ≤ω(h)− 〈A∗Dϕ(t, x), x〉 − f(x)V (t, x) +∂

∂tV (t, x)

+1

h

∫ t+h

t

Hτhu

(Dϕ(t, x) +Dg(x), D2ϕ(t, x) +D2g(x)

)du.

The conclusion is reached by first taking the supremum over all τ ∈ Σ inside the integraland then letting h go to 0.

5 Existence of solutions: F is weakly continuous

Throughout this section we assume that (B) holds and that Σ is a compact subset of Hd.Our goal is to show that the value function can be approximated uniformly on boundedsets by value functions of finite dimensional problems.

The operator A∗ is maximal monotone, therefore I + A∗ : D(A∗) → H is an isomor-phism when D(A∗) is equipped with the graph norm. Let V1 ⊂ V2 ⊂ ... ⊂ D(A∗) be finitedimensional subspaces such that D =

⋃∞n=1 Vn is a dense subset of D(A∗). D is also dense

in H . Let Pn be the orthogonal projection in H onto Vn. Define Aλ,n = PnA(I+λA)−1Pn,and consider the following stochastic differential equations

drλ,ns = (−Aλ,nr

λ,ns + Pnc(τs))ds+ Pnτs · dWs s ∈ (t, T ]

rλ,nt = x ∈ Hn.

(SEλ,n)

19

The solutions rλ,ns stay in Hn and since the operators Aλ,n are monotone the following

moment estimatesup

t≤s≤TE‖rλ,n

s ‖2p ≤ CT,p(1 + ‖x‖2p) for p ≥ 1 (47)

holds for some constant CT,p independent of both λ and n. Moreover for x ∈ Hn and‖x‖ ≤ r there is a constant Cr,T such that

E‖rλ,ns − x‖2 ≤ Cr,T (1 + λ−1)(s− t). (48)

The proofs of (47) and (48) follow standard arguments (see for instance [15]). The fact thatthe constant in (47) is independent of both λ and n is a consequence of the monotonicityof Aλ,n.

Let now Xλ,ns , Y λ,n

s denote the solutions of (SEλ,n) with initial data Xλ,nt = x, Y λ,n

n = yrespectively. Then

Xλ,ns − Y λ,n

s = e−(s−t)Aλ,n(x− y).

Therefore

supt≤s≤T

E‖Xλ,ns − Y λ,n

s ‖2 = ‖Xλ,nt − Y λ,n

t ‖2 ≤ ‖x− y‖2. (49)

We now defineUλ,n(t, x) = sup

ν∈ΘE

ν(e−

∫ Tt f(rλ,n

s )dsF (rλ,nT )).

Using (47), (48), (49), and the arguments of the proof of Proposition 4.1 we can concludethat there exists a modulus ρλ,r (independent of n) such that

|Uλ,n(t, x)− Uλ,n(s, x)| ≤ ρλ,r(|s− t|) ∀ x ∈ Hn, ‖x‖ ≤ r, 0 ≤ s, t ≤ T. (50)

The continuity in the x variable is studied in the proposition below. Let B be a compact,self-adjoint, positive operator such that B commutes with Pn for every n. Then the

metric (x, y) → ‖x − y‖B = 〈x− y, B(x− y)〉12 is equivalent on closed balls of H to the

one inherited from the weak topology.

Proposition 5.1. Let (B) hold. Let K be a positive number such that

|F (x)| ≤ K ∀ x ∈ H.

Then there exists a local modulus ω independent of λ, n, and t ∈ [0, T ] such that

|Uλ,n(t, x)− Uλ,n(t, y)| ≤ ω(‖x− y‖B, R) ∀ x, y ∈ BR ∩Hn. (51)

Proof. If (51) is not satisfied then there exist sequences λk, nk, points xk, yk ∈ BR∩Hn

and tk ∈ [0, T ] such that xk − yk converges weakly to 0, tk converges to some t, and

|Uλk,nk(tk, xk)− Uλk ,nk

(tk, yk)| ≥ ε0 for some ε0 > 0.

20

This implies that

supν∈Θ

(E

ν(|F (Xλk,nk

T )− F (Y λk,nkT )|)+KE

ν∣∣e− ∫ T

tkf(X

λk,nks )ds − e

− ∫ Ttk

f(Yλk,nk

s )ds∣∣) ≥ ε0,

where Xλk,nks , Y λk,nk

s are the solutions of (SEλk,nk) with the initial data Xλk,nk

tk= xk,

Y λk,nktk

= yk. Therefore, using the fact that f is nonnegative, we have for some stochasticbases νk that either

Eνk |F (Xλk,nk

T )− F (Y λk,nkT )| ≥ ε0

4(52)

or

Eνk∣∣ ∫ T

tk

(f(Xλk,nk

s )− f(Y λk,nks )

)ds∣∣ ≥ ε0

4K. (53)

For s fixed in (t, T ], k large enough, and ε > 0 we then have

Eνk(|f(Xλk,nk

s )− f(Y λk,nks )|) ≤ ωf

(‖Xλk,nks − Y λk,nk

s ‖B;1

ε

)+(P νk(‖Xλk,nk

s ‖ > 1

ε)

+ P νk(‖Y λk ,nks ‖ > 1

ε)) 1

2(E

νk(‖Xλk,nks ‖2 + ‖Y λk,nk

s ‖2)) 1

2 ,

where ωf is the local modulus of f in the B (weak) topology. It then follows from (47)that

Eνk(|f(Xλk,nk

s )− f(Y λk,nks )|) ≤ωf

(‖Xλk,nks − Y λk,nk

s ‖B;1

ε

)+ 2εC(1 +R2)

for some constant C. We can then select an appropriate constant Kε such that

Eνk(|f(Xλk,nk

s )− f(Y λk,nks )|) ≤ Kε‖Xλk,nk

s − Y λk,nks ‖B + ε+ 2εC(1 +R2).

Therefore

∫ T

tk

Eνk(|f(Xλk,nk

s )− f(Y λk,nks )|)ds ≤Kε

∫ T

tk

‖e−(s−tk)Aλk,nk (xnk− ynk

)‖Bds

+ 3εCT (1 +R2).

(54)

(Note that Xλk,nks − Y λk,nk

s is deterministic). We now want to let k → +∞ and ε→ 0 inthat order in (54). We need a lemma below.

Lemma 5.1. e−(s−tk)A∗

λk,nk converges to e−(s−t)A∗in the strong operator topology.

21

Proof of lemma. For every k, A∗λk,nk

is maximal monotone and thus −A∗λk ,nk

gener-ates a semi-group of contractions. For every x in D, A∗

λk,nkx converges to A∗x as k →∞,

and also (I+A∗)(D) is dense in H . Using one of the variants of the Trotter-Kato theorem

(see for example [34], Chapter 3, Theorem 4.5) e−uA∗

λk,nk converges to e−uA∗strongly and

the convergence is uniform for u in bounded intervals.

Therefore e−(s−tk)Aλk,nk (xnk− ynk

) 0 as k →∞ and since B is compact this impliesthat ‖e−(s−tk)Aλk,nk (xnk

− ynk)‖B → 0. Thus

limk→∞

Eνk∣∣ ∫ T

tk

(f(Xλ,n

k,s )− f(Y λ,nk,s )

)ds∣∣ = 0

which contradicts (53). The proof that (52) cannot hold is entirely similar. This concludesthe proof.

It is well known (see for instance [15]) that Uλ,n is a solution of the problem

∂

∂tUλ,n(t, x)− 〈Aλ,nx,DUλ,n(t, x)〉 − f(x)Uλ,n(t, x)

+Hn(DUλ,n(t, x), D2Uλ,n(t, x)) = 0,

Uλ,n(T, x) = F (x), (t, x) ∈ (0, T )× Vn

in the viscosity sense, where

Hn(p,X) = supτ∈Σ

〈p, Pnc(τ)〉 +

d∑k=1

〈XPnτk, Pnτk〉

for every p ∈ Vn and every symmetric n×n matrix X. The function Uλ,n can be extendedto H by Vλ,n(t, x) = Uλ,n(t, Pnx) which then is a viscosity solution of the equation

∂

∂tVλ,n(t, x)− 〈Aλ,nx,DVλ,n(t, x)〉

− f(Pnx)Uλ,n(t, x) +Hn(PnDUλ,n(t, x), PnD2Uλ,n(t, x)Pn) = 0,

Uλ,n(T, x) = F (Pnx), (t, x) ∈ (0, T )×H

(see [38]). For λ fixed, using (50), (51), and the Arzela-Ascoli theorem, we can concludethat Vλ,n converges uniformly on bounded sets of [0, T ]×H to a function Vλ that satisfies

|Vλ(t, x)− Vλ(s, x)| ≤ ρλ,r(|s− t|) ∀ x ∈ H, ‖x‖ ≤ r, 0 ≤ s, t ≤ T, (55)

|Vλ(t, x)− Vλ(t, y)| ≤ ω(‖x− y‖B, R) ∀ x, y ∈ BR. (56)

22

Moreover as in [38], Vλ is a solution in the viscosity sense of the equation

∂Vλ

∂t(t, x)− 〈Aλx,DVλ(t, x)〉 − x+(0)V (t, x)

+H(DVλ(t, x), D2Vλ(t, x)) = 0,

Vλ(T, x) = F (x) (t, x) ∈ (0, T )×H.

(CPλ)

In fact the modulus of continuity in (55) can be chosen uniformly in λ. To see this weproceed exactly as in [9] and [38]. We fix x in D(A) and t in (0, T ]. Let R > 0 be suchthat ‖x‖+ ‖Ax‖ ≤ R. Define

gε(t) = maxy∈H

Vλ(t, y)− 1

2ε‖x− y‖2 − µ(y)

,

where µ(x) = (1 + ‖x‖2)1/2 is the cut-off function from Section 3. It is clear that themaximum is achieved at some point y that remains in BR for some R independent ofε and that lim

ε↓012ε‖x − y‖2 = 0. Note that gε is continuous. Let ϕ be a continuously

differentiable function on (0, T ) such that gε − ϕ has a maximum at t0 ∈ (0, T ). Thisimplies that

(t, y) → Vλ(t, y)− 1

2ε‖x− y‖2 − µ(y)− ϕ(t)

has a maximum at some point (t0, y0). Using that Vλ is a viscosity solution of (CPλ) andthe monotonicity of Aλ we get

−ϕ′(t0) ≤ −〈Aλx,y0 − x

ε〉 − f(y0)Vλ(t0, y0)−H( y0 − x

ε+Dµ(y0), D

2µ(y0) +I

ε

).

It then follows that there exists a constant CR,ε such that

−g′ε(t) ≤ CR,ε

in the viscosity sense. Therefore we have

gε(t− h) ≤ CR,εh+ gε(t) ∀ t ∈ (0, T ), h > 0 such that t− h ∈ [0, T ].

this implies that

Vλ(t− h, x)− Vλ(t, x) ≤ CR,εh + ‖x− y‖+ ω(‖x− y‖, R)− ‖x− y‖2

2ε

≤ CR,εh + supr>0

r + ω(r, R)− r2

2ε

≤ CR,εh+ σ(ε),

where σ(ε) → 0 as ε→ 0. The reverse inequality can be obtained in a similar way. As in[9] this is enough to claim the existence of a modulus of continuity ρR such that

|Vλ(t, x)− Vλ(s, x)| ≤ ρR(|s− t|) ∀ x ∈ H, ‖x‖ ≤ R, 0 ≤ s, t ≤ T. (57)

Once again, using (56), (57), and the Arzela-Ascoli theorem, the Vλ converge uniformlyon bounded set of [0, T ]×H to a function V which is a viscosity solution of (CP ) in thesense of Definition 2.2. We refer the reader to [38] for the details of this.

23

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