# Malliavin Differentiability of Solutions of SPDEs with Lévy White Noise

Authors: Raluca M. Balan; Cheikh B. Ndongo
Journal: International Journal of Stochastic Analysis
 (2017)

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## Abstract

We consider a stochastic partial differential equation (SPDE) driven by a Lévy white noise, with Lipschitz multiplicative termσ. We prove that, under some conditions, this equation has a unique random field solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the Lévy white noise. If σ is affine, we prove that the solution is Malliavin differentiable and its Malliavin derivative satisfies a stochastic integral equation.


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## Body

## 1. Introduction

In this article, we consider the stochastic partial differential equation (SPDE):(1)Lut,x=σut,xL˙t,x,t∈0,T,x∈R,with some deterministic initial conditions, where L is a second-order differential operator on [0,T]×R, L˙ denotes the formal derivative of the Lévy white noise L (defined below), and the function σ:R→R is Lipschitz continuous.A processu=ut,x;t∈0,T,x∈R is called a (mild)solution of (1) if u is predictable and satisfies the following integral equation: (2)ut,x=wt,x+∫0t∫RGt-s,x-yσus,yLds,dy,where w is the solution of the deterministic equation Lu=0 with the same initial conditions as (1) and G is the Green function of the operator L.The study of SPDEs with Gaussian noise is a well-developed area of stochastic analysis, and the behaviour of random field solutions of such equations is well understood. We refer the reader to [1] for the original lecture notes which led to the development of this area and to [2, 3] for some recent advances. In particular, the probability laws of these solutions can be analyzed using techniques from Malliavin calculus, as described in [4, 5].On the other hand, there is a large literature dedicated to the study of stochastic differential equations (SDE) with Lévy noise, the monograph [6] containing a comprehensive account on this topic. One can develop also a Malliavin calculus for Lévy processes with finite variance, using an analogue of the Wiener chaos representation with respect to underlying Poisson random measure of the Lévy process. This method was developed in [7] with the same purpose of analyzing the probability law of the solution of an SDE driven by a finite variance Lévy noise. More recently, Malliavin calculus for Lévy processes with finite variance has been used in financial mathematics, the monograph [8] being a very readable introduction to this topic.There are two approaches to SPDEs in the literature. One is the random field approach which originates in John Walsh’s lecture notes [1]. When using this approach, the solution is viewed as a real-valued process which is indexed by time and space. The other approach is the infinite-dimensional approach, due to Da Prato and Zabczyk [9], according to which the solution is a process indexed by time only, which takes values in an infinite-dimensional Hilbert space. It is not always possible to compare the solutions obtained using the two approaches (see [10] for several results in this direction). SPDEs with Lévy noise were studied in the monograph [11], using the infinite-dimensional approach. In the present article, we use the random field approach for examining an SPDE driven by the finite variance Lévy noise introduced in [12], with the goal of studying the Malliavin differentiability of the solution. As mentioned above, this study can be useful for analyzing the probability law of the solution. We postpone this problem for future work.We begin by recalling from [12] the construction of the Lévy white noise L driving (1). We consider a Poisson random measure (PRM) N on the space U=[0,T]×R×R0 of intensity μ(dt,dx,dz)=dtdxνdz defined on a complete probability space Ω,F,P, where ν is a Lévy measure on R0=R∖0; that is, ν satisfies (3)∫R01∧z2νdz<∞.In addition, we assume that ν satisfies the following condition: (4)v≔∫R0z2νdz<∞.We denote byN^ the compensated PRM defined by N^(A)=N(A)-μ(A) for any A∈U with μ(A)<∞, where U is the class of Borel sets in U. We denote by Ft the σ-field generated by N0,s×B×Γ for all s∈0,t,B∈BbR, and Γ∈BbR0. We denote by BbR the class of bounded Borel sets in R and by BbR0 the class of Borel sets in R0 which are bounded away from 0.ALévy white noise with intensity measure ν is a collection L=LtB;t∈0,T,B∈BbR of zero-mean square-integrable random variables defined by (5)LtB=∫0t∫B∫R0zN^ds,dx,dz.These variables have the following properties:(i)
L0(B)=0 a.s. for all B∈Bb(R).(ii)
Lt(B1),…,LtBk are independent for any t>0 and for any disjoint sets B1,…,Bk∈Bb(R).(iii)
For any0<s≤t and for any B∈Bb(R), Lt(B)-Ls(B) is independent of Fs and has characteristic function(6)EeiuLtB-LsB=exp⁡t-sB∫R0eiuz-1-iuzνdz,u∈R.We denote by FtL the σ-field generated by Ls for all s∈0,t. For any h∈L20,T×R, we define the stochastic integral of h with respect to L: (7)Lh=∫0T∫Rht,xLdt,dx=∫0T∫R∫R0ht,xzN^dt,dx,dz.Using the same method as in Itô’s classical theory, this integral can be extended to random integrands, that is, to the class of predictable processesX={Xt,x;t∈[0,T],x∈R}, such that E∫0T∫R|X(t,x)|2dxdt<∞. The integral has the following isometry property:(8)E∫0T∫RXt,xLdt,dx2=vE∫0T∫RXt,x2dxdt.Recall that a process X=Xt,x;t≥0,x∈Rd ispredictable if it is measurable with respect to the predictable σ-field on Ω×R+×R, that is, the σ-field generated by elementary processes of the form (9)Xω,t,x=Yω1a,bt1Ax,ω∈Ω,t∈0,T,x∈R,where 0<a<b, Y is a bounded and FaL-measurable random variable, and A∈Bb(R).This article is organized as follows. In Section2, we introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure N^. In Section 3, we prove that, under a certain hypothesis, (1) has a unique solution. This hypothesis is verified in the case of the wave and heat equations. In Section 4, we examine the Malliavin differentiability of the solution, in the case when the function σ is affine. Finally, in the Appendix, we include a version of Gronwall’s lemma which is needed in the sequel.


## 2. Malliavin Calculus on the Poisson Space

In this section, we introduce the basic ingredients of Malliavin calculus with respect toN^, following very closely the approach presented in Chapters 10–12 of [8]. The difference compared to [8] is that our parameter space U has variables (t,x,z) instead of (t,z). For the sake of brevity, we do not include the proofs of the results presented in this section. These proofs can be found in Chapter 6 of the doctoral thesis [13] of the second author.We setH=L2U,U,μ and H⊗n=L2Un,Un,μn. We denote by H⊙ the set of all symmetric functions f∈H⊗n. We denote by HC,HC⊗n,HC⊙n the analogous spaces of C-valued functions.LetSn=u1,…,un∈Un;ui=ti,xi,ziwitht1<⋯<tn. For any measurable function f:Sn→R with (10)fL2Sn≔∫Snfu1,…,un2dμnu1,…,un<∞,we define the n-fold iterated integral of f with respect to N^ by (11)Jnf=∫0T∫R∫R0…∫0t2-∫R∫R0fu1,…,unN^du1…N^dun,where ui=ti,xi,zi. Then, EJnfJmg=0 for all n≠m and EJnf2=fL2(Sn)2.For anyf∈H⊙n, we defined themultiple integral of f with respect to N^ by In(f)=n!Jn(f). It follows that EInfImg=0 for all n≠m and (12)EInf2=n!fH⊗n2∀f∈H⊙n.If f∈HC⊙n with f=g+ih, we define Inf=Ing+iInh.LetLC2(Ω) be the set of C-valued square-integrable random variables defined on Ω,F,P. By Theorem 7 of [14], any FTL-measurable random variable F∈LC2Ω admits the chaos expansion (13)F=∑n≥0InfninLC2Ω,where fn∈H⊙n for all n≥1 and f0=E(F).The chaos expansion plays a crucial role in developing the Malliavin calculus with respect toN^. In particular, the Skorohod integrals with respect to N^ and L are defined as follows.Definition 1.
(a) LetX=Xu;u∈U be a square-integrable process such that Xu is FTL-measurable for any u∈U. For each u∈U, let X(u)=∑n≥0Infn·,u be the chaos expansion of X(u), with fn(·,u)∈H⊙n. One denotes by f~nu1,…,un,u the symmetrization of fn with respect to all n+1 variables. One says that X isSkorohod integrable with respect to N^ (and one writes X∈Dom⁡(δ)) if (14)∑n≥0EIn+1f~n2=∑n≥0n+1!f~nH⊙n+12<∞.In this case, one defines theSkorohod integral of X with respect to N^ by (15)δX=∫0T∫R∫R0Xt,x,zN^δt,δx,δz≔∑n≥0In+1f~n.
(b) LetY={Y(t,x);t∈[0,T],x∈R} be a square-integrable process such that Y(t,x) is FTL-measurable for any t∈[0,T] and x∈R. One says that Y isSkorohod integrable with respect to L (and one writes Y∈Dom⁡δL) if the process {Y(t,x)z;(t,x,z)∈U} is Skorohod integrable with respect to N^. In this case, one defines theSkorohod integral of Y with respect to L by (16)δLY=∫0T∫RYt,xLδt,δx≔∫0T∫R∫R0Yt,xzN^δt,δx,δz.
The following result shows that the Skorohod integral can be viewed as an extension of the Itô integral.Theorem 2.
(a) IfX=Xu;u∈U is a predictable process such that EXU2<∞, then X is Skorohod integrable with respect to N^ and (17)∫0T∫R∫R0Xt,x,zN^δt,δx,δz=∫0T∫R∫R0Xt,x,zN^dt,dx,dz.
(b) IfY=Yt,x;t∈0,T,x∈R is a predictable process such that E∫0T∫RYt,x2dxdt<∞, then Y is Skorohod integrable with respect to L and(18)∫0T∫RYt,xLδt,δx=∫0T∫RYt,xLdt,dx.We now introduce the definition of the Malliavin derivative.Definition 3.
LetF∈L2(Ω) be an FTL-measurable random variable with the chaos expansion F=∑n≥0In(fn) with fn∈H⊙n. One says that F isMalliavin differentiable with respect to N^ if (19)∑n≥1nn!fnH⊗n2<∞.In this case, one defines theMalliavin derivative of F with respect to N^ by (20)DuF=∑n≥1nIn-1fn·,u,u∈U.One denotes by D1,2 the space of Malliavin differentiable random variables with respect to N^.Note thatEDFH2=∑n≥1nn!fnH⊗n2<∞.Theorem 4 (closability of Malliavin derivative).
LetFnn≥1⊂D1,2 and F∈L2(Ω) such that Fn→F in L2Ω and DFnn≥1 converges in L2(Ω;H). Then, F∈D1,2 and DFn→DF in L2Ω;H.Typical examples of Malliavin differentiable random variables are exponentials of stochastic integrals: for anyh∈L2([0,T]×R), (21)Dt,x,zeLh=eLheht,xz-1.Moreover, the set DE1,2 of linear combinations of random variables of the form eL(h) with h∈L20,T×R is dense in D1,2.The following result shows that the Malliavin derivative is a difference operator with respect toN^, not a differential operator.Theorem 5 (chain rule).
For anyF∈D1,2 and any continuous function g:R→R such that gF∈L2Ω and gF+DF-gF∈L2Ω;H, g(F)∈D1,2 and (22)DgF=gF+DF-gFinL2Ω;H.Similar to the Gaussian case, we have the following results.Theorem 6 (duality formula).
IfF∈D1,2 and X∈Dom⁡(δ), then (23)EF∫0T∫R∫R0Xt,x,zN^δt,δx,δz=E∫0T∫R∫R0Xt,x,zDt,x,zFνdzdxdt.Theorem 7 (fundamental theorem of calculus).
LetX=Xs,y,ζ;s∈0,T,y∈R,ζ∈R0 be a process which satisfies the following conditions:(i)
Xs,y,ζ∈D1,2 for any s,y,ζ∈U.(ii)
E∫0T∫R∫R0Xs,y,ζ2νdzdyds<∞.(iii)
Dt,x,zXs,y,ζ;s,y,ζ∈U∈Dom⁡(δ) for any t,x,z∈U.(iv)
δDt,x,zX;t,x,z∈U∈L2Ω;H.
Then,X∈Dom⁡(δ), δ(X)∈D1,2 and D[δ(X)]=X+δ(DX); that is, (24)Dt,x,z∫0T∫R∫R0Xs,y,ζN^δs,δy,δζ=Xt,x,z+∫0T∫R∫R0Dt,x,zXs,y,ζN^δs,δy,δζinL2Ω;H.As an immediate consequence of the previous theorem, we obtain the following result.Theorem 8.
LetY=Ys,y;s∈0,T,y∈R be a process which satisfies the following conditions:(i)
Ys,y∈D1,2 for all s∈[0,T] and y∈R.(ii)
E∫0T∫RYs,y2dyds<∞.(iii)
Dt,x,zYs,y;s∈0,T,y∈R∈Dom⁡(δL) for any (t,x,z)∈U.(iv)
E∫0T∫R∫R0∫0T∫RDt,x,zYs,yLδs,δy2νdzdxdt<∞.Then, Y∈Dom⁡(δL), δL(Y)∈D1,2 and the following relation holds in L2(Ω;H): (25)Dt,x,zδLY=Yt,xz+∫0T∫R∫R0Dt,x,zYs,yLδs,δy.


## 3. Existence of Solution

In this section, we show that (1) has a unique solution.We recall thatw is the solution of the homogeneous equation Lu=0 with the same initial conditions as (1), and G is the Green function of the operator L on R+×R. We assume that, for any t∈0,T, Gt,·∈L1R and we denote by FGt,· its Fourier transform: (26)FGt,·ξ=∫Re-iξxGt,xdx.We suppose that the following hypotheses hold.Hypothesis H1. w is continuous and uniformly bounded on [0,T]×R.Hypothesis H2. (a) ∫0T∫RG2t,xdxdt<∞; (b) the function t↦FG(t,·)(ξ) is continuous on [0,T], for any ξ∈Rd; (c) there exist ε>0 and a nonnegative function kt(·) such that (27)FGt+h,·ξ-FGt,·ξ≤ktξfor any t∈[0,T] and h∈[0,ε], and ∫0T∫Rkt(ξ)2dξdt<∞.Sinceσ is a Lipschitz continuous function, there exists a constant Cσ>0 such that, for any x,y∈R,(28)σx-σy≤Cσx-y.In particular, for any x∈R,(29)σx≤Dσ1+x,where Dσ=max⁡Cσ,σ0.The following theorem is an extension of Theorem1.1.(a) of [15] to an arbitrary operator L.Theorem 9.
Equation (1) has a unique solution u=ut,x;t∈0,T,x∈R which is L2(Ω)-continuous and satisfies (30)supt,x∈0,T×R⁡Eut,x2<∞.Proof.
Existence. We use the same argument as in the proof of Theorem 13 of [16]. We denote by (un)n≥0 the sequence of Picard iterations defined by u0(t,x)=w(t,x) and(31)un+1t,x=wt,x+∫0t∫RGt-s,x-yσuns,yLds,dy,n≥0.By induction on n, it can be proved that the following properties hold:(P)
(i)
unt,x is well defined for any t,x∈0,T×R.(ii)
Kn≔sup(t,x)∈[0,T]×R⁡Eunt,x2<∞.(iii)
t,x↦unt,x is L2Ω-continuous on 0,T×R.(iv)
unt,x is Ft-measurable for any t∈0,T and x∈R.Hypotheses (H1) and (H2) are needed for the proof of property (iii). From properties (iii) and (iv), it follows that un has a predictable modification, denoted also by un. This modification is used in definition (31) of un+1(t,x). Using the isometry property (8) of the stochastic integral and (28), we have(32)Eun+1t,x-unt,x2=vE∫0t∫RG2t-s,x-yσuns,y-σun-1s,y2dyds≤vCσ2∫0t∫RG2t-s,x-yEuns,y-un-1s,y2dyds≤vCσ2∫0tHns∫RG2t-s,x-ydyds,where Hn(t)=supx∈R⁡E|un(t,x)-un-1(t,x)|2. For any t∈[0,T], we denote(33)Jt=∫RG2t,xdx.Taking the supremum over x∈R in the previous inequality, we obtain that (34)Hn+1t≤vCσ2∫0tHnsJt-sds,for any t∈[0,T] and n≥0. By applying Lemma A.1 (the Appendix) with Cn=0 and p=2, we infer that(35)∑n≥0supt∈0,T⁡Hnt1/2<∞.This shows that the sequence (un)n≥0 converges in L2(Ω) to a random variable u(t,x), uniformly in [0,T]×R; that is,(36)supt,x∈0,T×R⁡Eunt,x-ut,x2⟶0.To see that u is a solution of (1), we take the limit in L2(Ω) as n→∞ in (31). In particular, this argument shows that(37)K≔supn≥1⁡supt,x∈0,T×R⁡Eunt,x2<∞.Uniqueness. Let H(t)=supx∈R⁡Eut,x-u′t,x2, where u and u′ are two solutions of (1). A similar argument as above shows that (38)Ht≤vCσ2∫0tHsJt-sds,for any t∈[0,T]. By Gronwall’s lemma, H(t)=0 for all t∈[0,T].Example 10 (wave equation).
IfLu=∂u/∂2t-∂u/∂x2 for t∈[0,T] and x∈R, then G(t,x)=1/21{|x|≤t}. Hypothesis (H2) holds since(39)FGt,·ξ=sin⁡tξξ.Example 11 (heat equation).
IfLu=∂u/∂t-1/2∂u/∂x2 for t∈0,T and x∈R, then Gt,x=2πt-1/2exp⁡-|x|2/2t. Hypothesis (H2) holds since (40)FGt,·ξ=exp⁡-tξ22.


## 4. Malliavin Differentiability of the Solution

In this section, we show that the solution of (1) is Malliavin differentiable and its Malliavin derivative satisfies a certain integral equation. For this, we assume that the function σ is affine.Our first result shows that the sequence of Picard iterations is Malliavin differentiable with respect toN^ and the corresponding sequence of Malliavin derivatives is uniformly bounded in L2Ω;H.Lemma 12.
Assume thatσ is an arbitrary Lipschitz function. Let unn≥0 be the sequence of Picard iterations defined by (31). Then, unt,x∈D1,2 for any t,x∈0,T×R and n≥0, and (41)A≔supn≥0⁡supt,x∈0,T×R⁡EDunt,xH2<∞.Proof.
Step
1. We prove that the following property holds for any n≥0:Qunt,x∈D1,2for  anyt,x∈0,T×R,An≔supt,x∈0,T×R⁡EDunt,xH2<∞.For this, we use an induction argument on n. Property Q is clear for n=0. We assume that it holds for n and we prove that it holds for n+1.
By the definition ofun+1 and the fact that the Itô integral coincides with the Skorohod integral if the integrand is predictable, it follows that, for any t,x∈0,T×R, (42)un+1t,x=wt,x+∫0t∫RGt-s,x-yσuns,yLδs,δy.
We fix(t,x)∈[0,T]×R. We apply the fundamental theorem of calculus for the Skorohod integral with respect to L (Theorem 8) to the process: (43)Ys,y=Gt-s,x-yσuns,y10,ts,s∈0,T,y∈R.We need to check that Y satisfies the hypotheses of this theorem. To check that Y satisfies (i), we apply the chain rule (Theorem 5) to F=un(s,y) and g=σ. Note that, for any (s,y)∈[0,T]×R,(44)Eσuns,y2≤2Dσ21+Euns,y2≤2Dσ21+Kn<∞,(45)E∫0T∫R∫R0σuns,y+Dr,ξ,zuns,y-σuns,y2νdzdξdr≤Cσ2E∫0T∫R∫R0Dr,ξ,zuns,y2νdzdξdr≤Cσ2An<∞,by the induction hypothesis. We conclude that Y(s,y)∈D1,2 and(46)Dr,ξ,zYs,y=Gt-s,x-yσuns,y+Dr,ξ,zuns,y-σuns,y10,ts.
We note thatY satisfies hypothesis (ii) since, by (44), (47)E∫0T∫RYs,y2dyds≤2Dσ21+Kn∫0t∫RG2t-s,x-ydyds<∞.
To check thatY satisfies hypothesis (iii), i.e., the process Dr,ξ,zYs,y;s∈0,T,y∈R is Skorohod integrable with respect to L for any r,ξ,z∈U, it suffices to show that this process is Itô integrable with respect to L. Note that Dr,ξ,zun(s,y)=0 if r>s and it is Fs-measurable if r≤s. Hence, the process Dr,ξ,zYs,y;s∈0,T,y∈R is predictable. By (46) and (28), (48)E∫0T∫RDr,ξ,zYs,y2dyds≤Cσ2E∫0t∫RG2t-s,x-yDr,ξ,zuns,y2dyds,and, hence,(49)∫0T∫R∫R0E∫0T∫RDr,ξ,zYs,y2dydsνdzdξdr≤Cσ2∫0t∫RG2t-s,x-yEDuns,yH2dyds≤Cσ2An∫0t∫RG2t-s,x-ydyds<∞.This proves that E∫0T∫RDr,ξ,zYs,y2dyds<∞ for almost all r,ξ,z∈[0,T]×R×R0. By Theorem 2 (b), Dr,ξ,zYs,y;s∈0,T,y∈R is Skorohod integrable with respect to L and(50)∫0T∫RDr,ξ,zYs,yLδs,δy=∫0T∫RDr,ξ,zYs,yLds,dy.
Finally,Y satisfies hypothesis (iv) since, by (50) and the isometry properties (8) and (49), we have (51)E∫0T∫R∫R0∫0T∫R∫R0Dr,ξ,zYs,yLδs,δy2νdzdξdr=E∫0T∫R∫R0∫0T∫R∫R0Dr,ξ,zYs,yLds,dy2νdzdξdr=v∫0T∫R∫R0E∫0T∫R∫R0Dr,ξ,zYs,y2dydsνdzdξdr<∞.
By Theorem8, we infer that Y∈Dom⁡(δL), δLY∈D1,2, and(52)Dr,ξ,zδLY=Yr,ξz+∫0t∫RDr,ξ,zYs,yLδs,δy.Since un+1(t,x)=w(t,x)+δL(Y), this means that un+1(t,x)∈D1,2. Using (50) and (46), we can rewrite relation (52) as follows:(53)Dr,ξ,zun+1t,x=Gt-r,x-ξσunr,ξz+∫0t∫RGt-s,x-yσuns,y+Dr,ξ,zuns,y-σuns,yLds,dy.
It remains to prove that(54)An+1=supt,x∈0,T×R⁡EDun+1t,xH2<∞.Using (53), the isometry property (8), relation (44), and the fact that σ is Lipschitz, we see that (55)EDr,ξ,zun+1t,x2≤2z2G2t-r,x-ξEσunr,ξ2+2vE∫0t∫RG2t-s,x-yσuns,y+Dr,ξ,zuns,y-σuns,y2dyds≤4z2Dσ21+KnG2t-r,x-ξ+2vCσ2E∫0t∫RG2t-s,x-yDr,ξ,zuns,y2dyds.We perform integration with respect to drdξνdz on [0,T]×R×R0. We denote(56)νt=∫0t∫RG2s,ydyds.We obtain(57)EDun+1t,xH2≤4vDσ21+Knνt+2vCσ2∫0t∫RG2t-s,x-yEDuns,yH2dyds≤4vDσ21+Knνt+2vCσ2Anνt.Relation (54) follows taking the supremum over t,x∈[0,T]×R.
Step
2. We prove that supn≥1⁡An<∞. By (57), we have (58)EDun+1t,xH2≤4vDσ21+Knνt+2vCσ2∫0tVnsJt-sds,where Vn(t)=supx∈REDunt,xH2 and J(t) is given by (33). This shows that (59)Vn+1t≤4vDσ2νT1+K+2vCσ2∫0tVnsJt-sds,where K is given by (37). By Lemma 15 of [16], supn≥1⁡supt∈0,T⁡Vnt<∞.We are now ready to state the main result of the present article.Theorem 13.
Assume thatσ is an affine function; that is, σ(x)=ax+b for some a,b∈R. If u is the solution of (1), then, for any t∈[0,T] and x∈R, (60)ut,x∈D1,2and the following relation holds in L2(Ω;H) (and hence, almost surely, for μ-almost all (r,ξ,z)∈U):(61)Dr,ξ,zut,x=Gt-r,x-ξσur,ξz+∫0t∫RGt-s,x-yσus,y+Dr,ξ,zus,y-σus,yLds,dy.Proof.
Step
1. For any t∈[0,T] and n≥0, let (62)Mnt=supx∈R⁡EDunt,x-Dun-1t,xH2.Note that, by Lemma 12, un(t,x)∈D1,2 for any (t,x)∈[0,T]×R and n≥0.
Fix(r,ξ,z)∈U. We write relation (53) for Dr,ξ,zun+1(t,x) and Dr,ξ,zun(t,x). We take the difference between these two equations. We obtain(63)Dr,ξ,zun+1t,x-Dr,ξ,zunt,x=Gt-r,x-ξσunr,ξ-σun-1r,ξz+∫0t∫RGt-s,x-yσuns,y+Dr,ξ,zuns,y-σuns,y-σun-1s,y+Dr,ξ,zun-1s,y-σun-1s,yLds,dy.
At this point, we use the assumption thatσ is the affine function σ(x)=ax+b. (An explanation of why this argument does not work in the general case is given in Remark 14.) In this case, relation (63) has the following simplified expression: (64)Dr,ξ,zun+1t,x-Dr,ξ,zunt,x=aGt-r,x-ξunr,ξ-un-1r,ξz+a∫0t∫RGt-s,x-yDr,ξ,zuns,y-Dr,ξ,zun-1s,yLds,dy.Using Itô’s isometry and the inequality (a+b)2≤2(a2+b2), we obtain (65)EDr,ξ,zun+1t,x-Dr,ξ,zunt,x2≤2a2z2G2t-r,x-ξbn2+2a2vE∫0t∫RG2t-s,x-yDr,ξ,zuns,y-Dr,ξ,zun-1s,y2dyds,where bn2=sup(s,y)∈[0,T]×R⁡Euns,y-un-1s,y2. Note that both sides of the previous inequality are zero if r>t. Taking the integral with respect to drdξνdz on 0,T×R×R0, we obtain (66)EDun+1t,x-Dunt,xH2≤2a2vνtbn2+2a2vE∫0t∫RG2t-s,x-yEDuns,y-Dun-1s,yH2dyds,where νt is given by (56). Recalling the definition of Mn(t), we infer that (67)Mn+1t≤Cn+2a2v∫0tMnsJt-sds,where Cn=2a2vνtbn2 and the function J is given by (33). By relation (35), we know that ∑n≥1bn<∞, which means that ∑n≥1Cn1/2<∞. By Lemma A.1 (the Appendix), we conclude that (68)∑n≥1supt≤T⁡Mnt1/2<∞.Hence, the sequence {Dun(t,x)}n≥1 converges in L2(Ω;H) to a variable U(t,x), uniformly in (t,x)∈[0,T]×R; that is,(69)supt,x∈0,T×R⁡EDunt,x-Ut,xH2⟶0.
Step
2. We fix (t,x)∈[0,T]×R. By (36), un(t,x)→u(t,x) in L2(Ω). By Step 1, {Dun(t,x)}n≥1 converges in L2(Ω;H). We apply Theorem 4 to the variables Fn=un(t,x) and F=u(t,x). We infer that u(t,x)∈D1,2 and Dun(t,x)→Du(t,x) in L2(Ω;H). Combining this with (69), we obtain (70)supt,x∈0,T×R⁡EDunt,x-Dut,xH2⟶0.Relation (61) follows by taking the limit in L2(Ω;H) as n→∞ in (53).Remark 14.
Unfortunately, we were not able to extend Theorem13 to an arbitrary Lipschitz function σ. To see where the difficulty comes from, recall that we need to prove that Dun(t,x)n≥1 converges in L2(Ω;H), and the difference Dr,ξ,zun+1(t,x)-Dr,ξ,zun(t,x) is given by (63). For an arbitrary Lipschitz function σ, by relation (28), we have (71)σuns,y+Dr,ξ,zuns,y-σun-1s,y+Dr,ξ,zun-1s,y≤Cσuns,y+Dr,ξ,zuns,y-un-1s,y+Dr,ξ,zun-1s,y≤Cσuns,y-un-1s,y+Dr,ξ,zuns,y-Dr,ξ,zun-1s,y.
Using (63), the isometry property (8), the inequality (a+b)2≤2(a2+b2), and the previous inequality, we have (72)EDr,ξ,zun+1t,x-Dr,ξ,zunt,x2≤2z2Cσ2G2t-r,x-ξEunr,ξ-un-1r,ξ2+4vCσ2∫0t∫RG2t-s,x-yEuns,y-un-1s,y2dyds+4vCσ2∫0t∫RG2t-s,x-yEDr,ξ,zuns,y-Dr,ξ,zun-1s,ydyds.The problem is that the second term on the right-hand side of the inequality above does not depend on (r,ξ,z) and hence its integral with respect to drdξν(dz) on [0,T]×R×R0 is equal to ∞.



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*Source: 9693153-2017-03-12.xml*