# Convergence Theorems for Operators Sequences on Functionals of Discrete-Time Normal Martingales

Authors: Jinshu Chen
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We aim to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingalesM. We first apply the 2D-Fock transform for operators from the testing functional space S(M) to the generalized functional space S⁎(M) and obtain a necessary and sufficient condition for such operators sequences to be strongly convergent. We then discuss the integration of these operator-valued functions. Finally, we apply the results obtained here and establish the existence and uniqueness of solution to quantum stochastic differential equations in terms of operators acting on functionals of discrete-time normal martingales M. And also we prove the continuity and continuous dependence on initial values of the solution.


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## Body

## 1. Introduction

White noise analysis created by Hida [1] is essentially a branch of infinite-dimensional calculus on generalized functionals of Brownian motion, which connected with the applications to the study of random processes and stochastic differential equations. Ito’s theory is based on the notion of the Wiener measure in the space of continuous functions, interpreted as the trajectories of Brownian motion. And considering the space of slowly growing Schwartz distributions as the space of trajectories of white noise, Hida has constructed a new theory of white noise functionals [2–4], which proved to be useful in the studying of stochastic differential equations and was further developed in [5, 6] and other papers. On the other hand, many problems in mathematical physics and financial mathematics lead to the necessity of studying stochastic differential equations with operator coefficients in infinite-dimensional spaces. In [7], the authors generalize the classical Ito theory to this case.In 1998, Ito introduced his analysis of generalized Poisson functionals in paper [8], which can be viewed as a theory of infinite-dimensional calculus on generalized functionals of Poisson martingale. As we know, both Brownian motion and Poisson martingale are continuous-time normal martingales. And other continuous-time processes theories of white noise analysis appeared in [9–12].Discrete-time normal martingales [13] also play an important role in many theoretical and applied fields. For example, the classical random walk is just such a discrete-time normal martingale [14, 15]. It would then be interesting to develop a theory of infinite-dimensional calculus on generalized functionals of discrete-time normal martingales.LetM=Mnn∈N be a discrete-time normal martingale satisfying some mild conditions. In paper [16], the authors constructed the testing functional space S(M) and generalized functional space S∗(M) of M by using a specific orthonormal basis for square integrable functionals of M and also characterized these functionals via a transform acting on them.It is well known that operators on functional spaces play a fundamental role in quantum mechanics. In paper [17], the authors introduce a transform, called 2D-Fock transform, for operators from the testing functional space S(M) to the generalized functional space S∗(M) of M and characterize continuous linear operators from S(M) to S∗(M). In this paper, we apply the 2D-Fock transform to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales M. We obtain a necessary and sufficient condition for such sequence to be strongly convergent. And we give a criterion for checking whether such operator-valued functions are Bochner-integrable.Finally, as application of the results obtained here, we investigate the following nonlinear quantum stochastic differential equation:(1)dTtdt=Xt,Tt+Yt,Tt∗Nt,t∈R+T0=J,where X,Y:R×L[S(M),S∗(M)]→L[S(M),S∗(M)] are two maps, J∈L[S(M),S∗(M)] is the initial value, and ∗ stands for the convolution of operators. (N(t))t∈R+ is a given L[S(M),S∗(M)]-valued quantum stochastic process which plays the role of quantum noises, and the solution (T(t))t∈R+ will be a L[S(M),S∗(M)]-valued quantum stochastic process.Equation (1) can describe the evolution of a quantum system at a level of operators in the presence of quantum noise. We will establish the existence and uniqueness of solution to equation (1). And also we prove the continuity and continuous dependence on initial values of the solution.The paper is organized as follows. In Section2, we briefly recall the construction and characterization of continuous linear operators from S(M) to S∗(M), which were made in recent paper [17]. In Section 3, we apply the 2D-Fock transform to investigate the convergence of operators sequences acting on functionals of discrete-time normal martingales M. We first obtain a necessary and sufficient condition for such sequence to be strongly convergent. And then we give a criterion for checking whether such operator-valued functions are Bochner-integrable. In Section 4, we apply the results obtained here and establish the existence and uniqueness of solution to quantum stochastic differential equations (1) and prove the continuity and continuous dependence on initial values of the solution.Notation and Conventions.Throughout the paper, N designates the set of all nonnegative integers and Γ denotes the finite power set of N; namely,(2)Γ=σ∣σ⊂N,#σ<∞,where #(σ) means the cardinality of σ as a set. In addition, we always assume that (Ω,F,P) is a given probability space with E denoting the expectation with respect to P. We denote by L2(Ω,F,P) the usual Hilbert space of square integrable complex-valued functions on (Ω,F,P) and use ·,· and · to mean its inner product and norm, respectively. By convention, 〈·,·〉 is conjugate-linear in its first argument and linear in its second argument.


## 2. Generalized Functionals of Discrete-Time Normal Martingales

Definition 1.
A stochastic processM=(Mn)n∈N on (Ω,F,P) is called a discrete-time normal martingale if it is square integrable and satisfies(i)
E[M0∣F-1]=0 and E[Mn∣Fn-1]=Mn-1 for n≥1;(ii)
E[M02∣F-1]=1 and E[Mn2∣Fn-1]=Mn-12+1 for n≥1,
whereF-1={∅,Ω}, Fn=σ(Mk;0≤k≤n) for n∈N, and E[·∣Fk] means the conditional expectation.LetM=(Mn)n∈N be a discrete-time normal martingale on (Ω,F,P). Then one can construct from M a process Z=(Zn)n∈N as(3)Z0=M0,Zn=Mn-Mn-1,n≥1.It can be verified that Z admits the following properties:(4)EZn∣Fn-1=0,EZn2∣Fn-1=1,n∈N.Thus Z can be viewed as a discrete-time noise, which we call the discrete-time normal noise associated with M.Lemma 2 (see [18, 19]).
LetZ=(Zn)n∈N be the discrete-time normal noise associated with M. Define Z∅=1, where ∅ denotes the empty set and(5)Zσ=∏i∈σZi,σ∈Γ,σ≠∅. Then {Zσ∣σ∈Γ} forms a countable orthonormal system in L2(Ω,F,P).LetF∞=σ(Mn;n∈N); the σ-field over Ω is generated by M. In the literature, F∞-measurable functions on Ω are also known as functionals of M. Thus elements of L2(Ω,F∞,P) can be called square integrable functionals of M.Definition 3.
The discrete-time normal martingaleM is said to have the chaotic representation property if the system {Zσ∣σ∈Γ} defined by (5) is total in L2(Ω,F∞,P).So, if the discrete-time normal martingaleM has the chaotic representation property, then the system {Zσ∣σ∈Γ} defined by (5) is actually an orthonormal basis for L2(Ω,F∞,P), which is a closed subspace of L2(Ω,F,P) as is known.And, for brevity, we useL2(M) to mean the space of square integrable functionals of M; namely,(6)L2M=L2Ω,F∞,P,which shares the same inner product and norm with L2(Ω,F,P), namely, 〈·,·〉 and ·. We note that {Zσ∣σ∈Γ} forms a countable orthonormal basis for L2(M).Lemma 4 (see [20]).
Letσ↦λσ be the N-valued function on Γ given by(7)λσ=∏k∈σk+1,σ≠∅,σ∈Γ;1,σ=∅,σ∈Γ. Then, for p>1, the positive term series ∑σ∈Γλσ-p converges and moreover(8)∑σ∈Γλσ-p≤exp∑k=1∞k-p<∞.Using theN-valued function defined by (7), we can construct a chain of Hilbert spaces consisting of functionals of M as follows. For p≥0, we define a norm ·p on L2(M) through(9)ξp2=∑σ∈Γλσ2pZσ,ξ2,ξ∈L2Mand put(10)SpM=ξ∈L2M∣ξp<∞. It is not hard to check that ·p is a Hilbert norm and Sp(M) becomes a Hilbert space with ·p.Lemma 5 (see [16]).
Forp≥0, one has {Zσ∣σ∈Γ}⊂Sp(M) and moreover the system {λσ-pZσ∣σ∈Γ} forms an orthonormal basis for Sp(M).It is easy to see thatλσ≥1 for all σ∈Γ. This implies that ·p≤·q and Sq(M)⊂Sp(M) whenever 0≤p≤q. Thus we actually get a chain of Hilbert spaces of functionals of M:(11)⋯⊂Sp+1M⊂SpM⊂⋯⊂S1M⊂S0M=L2M.We now put(12)SM=⋂p=0∞SpMand endow it with the topology generated by the norm sequence {·p}p≥0. Note that, for each p≥0, Sp(M) is just the completion of S(M) with respect to ·p. Thus S(M) is a countably-Hilbert space [21]. The next lemma, however, shows that S(M) even has a much better property.Lemma 6 (see [16]).
The spaceS(M) is a nuclear space; namely, for any p≥0, there exists q>p such that the inclusion mapping ipq:Sq(M)→Sp(M) defined by ipq(ξ)=ξ is a Hilbert-Schmidt operator.Forp≥0, we denote by Sp∗(M) the dual of Sp(M) and by ·-p the norm of Sp∗(M). Then Sp∗(M)⊂Sq∗(M) and ·-p≥·-q whenever 0≤p≤q. The lemma below is then an immediate consequence of the general theory of countably-Hilbert spaces (see [21]).Lemma 7 (see [16]).
LetS∗(M) be the dual of S(M) and endow it with the strong topology. Then(13)S∗M=⋃p=0∞Sp∗Mand moreover the inductive limit topology on S∗(M) given by space sequence {Sp∗(M)}p≥0 coincides with the strong topology.We mention that, by identifyingL2(M) with its dual, one comes to a Gel’fand triple,(14)SM⊂L2M⊂S∗M,which we refer to as the Gel’fand triple associated with M.Definition 8 (see [16]).
Elements ofS∗(M) are called generalized functionals of M, while elements of S(M) are called testing functionals of M.Throughout this paper, we denote byL the set of all continuous linear operators from S(M) to S∗(M); that is, L=L[S(M),S∗(M)].Definition 9 (see [17]).
For an operatorT∈L, its 2D-Fock transform is the function T^ on Γ×Γ given by(15)T^σ,τ=TZσ,Zτ,σ,τ∈Γ×Γ, where ·,· is the canonical bilinear form on S∗(M)×S(M).Much like generalized functionals ofM, continuous linear operators in L are also completely determined by their 2D-Fock transforms.Lemma 10 (see [17]).
LetT1, T2:S(M)→S∗(M) be continuous linear operators. Then T1=T2 if and only if T1^=T2^.The following lemma is known as the characterization theorem of operators inL through their 2D-Fock transforms.Lemma 11 (see [17]).
LetG be a function on Γ×Γ. Then G is the 2D-Fock transform of an element T in L if and only if it satisfies(16)Gσ,τ≤Cλσpλτp,σ,τ∈Γ×Γfor some constants C≥0 and p≥0. In that case, for q>p+1/2, one has(17)Tq≤C∑σ∈Γλσ-2q-pand in particular T takes values in Sq∗(M), where(18)Tp≡supξ≠0,ξ∈SMTξ-pξp, where p≥0.For two operatorsT1, T2∈L, their usual product T1T2 may not make sense. However, one can introduce a product of other type for them.Definition 12 (see [17]).
LetT1, T2∈L; then their convolution T1∗T2 is defined as(19)T1∗T2^σ,τ=T1^σ,τT2^σ,τ,σ,τ∈Γ×Γ.It can be verified that(L,*) forms a commutative algebra with an involution and a unit.


## 3. Convergence Theorems for Operators inL

LetM=(Mn)n∈N be the same discrete-time normal martingale as described in Section 2. In the present section, we apply the 2D-Fock transform to establish convergence theorems for operators in L. Furthermore, we discuss the integration of these operator-valued functions.Definition 13.
A sequence(Tn)n≥1⊂L is called strongly convergent to T∈L, if, for any ξ∈S(M), one has Tn(ξ)→T(ξ) (in the strong topology of S∗(M)).The following theorem offers a necessary condition for operators sequences inL to be strongly convergent.Theorem 14.
Let(Tn)n≥1 be an operator sequence in L, and let T be operator in L. Then the sequence (Tn)n≥1 converges strongly to T if and only if(1)
Tn^(σ,τ)→T^(σ,τ) for all σ,τ∈Γ;(2)
there exist constantsC≥0 and p≥0 such that(20)supn≥1Tn^σ,τ≤Cλσpλτp,σ,τ∈Γ×Γ.Proof.
For the “only if” part, let(Tn)n≥1 converge strongly to T; then it is easy to see that(21)Tn^σ,τ=TnZσ,Zτ⟶TZσ,Zτ=T^σ,τ,σ,τ∈Γ.On the other hand, by Lemma 11, we know that, for each n, there exist Cn≥0 and p≥0 such that(22)Tn^σ,τ≤Cnλσpλτp,σ,τ∈Γ×Γ.Let C=supn≥1Cn, according to the principle of uniform boundedness, C<∞, which implies that(23)supn≥1Tn^σ,τ≤Cλσpλτp,σ,τ∈Γ×Γ.
For the “if” part, let(Tn)n≥1 satisfy conditions (1) and (2); by taking q>p+1/2 and using Lemma 11, we get(24)supn≥1Tnξ-q≤C∑σ∈Γλσ-2q-pξq,ξ∈SM,which means that (Tn)n≥1 is an operator sequence from Sq(M) to Sq∗(M) and it is uniformly bounded. On the other hand, {Zσ∣σ∈Γ} is total in Sq(M), and condition (1) equivalent to TnZσ,Zτ converges in C for any σ,τ∈Γ, so there exists a linear operator form Sq(M) to Sq∗(M) such that, for any ξ,η∈Sq(M), we have(25)Tnξ,η⟶Tξ,ηn⟶∞;thus T is a linear operator form S(M) to S∗(M), and, for any ξ∈S(M), Tn(ξ) converges weakly to T(ξ) in S∗(M). By Proposition 3.1 in paper [16], Tn(ξ) converges strongly to T(ξ) in S∗(M). So (Tn)n≥1 converges strongly to T.In the following, we discuss the integration ofL-valued mapping.Definition 15.
Let(E,E,ν) be a measure space and let T(·):E→L be a L-valued function; then T(·) is said to be strongly Bochner-integrable with respect to ν if generalized functional valued function T(·)ξ:E→S∗(M) is Bochner-integrable with respect to ν for any ξ∈S(M). In that case, a linear operator from S(M) to S∗(M),(26)ξ⟶∫ETωξdνω,is defined. We called it the Bochner integral of T· with respect to ν and it is denoted by ∫ET(ω)dν(ω).We note that the strong Bochner integral∫ET(ω)dν(ω) may not be an operator in L. The next theorem, however, provides sufficient conditions for T(·) to be strongly Bochner-integrable and its strong Bochner integral to be an operator in L.Theorem 16.
LetT(·):E→L be a L-valued mapping satisfying the following conditions:(1)
For anyσ,τ∈Γ, the function T(·)^(σ,τ):E→C is measurable.(2)
There existp≥0 and a nonnegative function C(ω)∈L1(E,ν), such that, for ν-a.e., ω∈E, it holds that(27)Tω^σ,τ≤Cωλσpλτpσ,τ∈Γ.Then T(·) is strongly Bochner-integrable with respect to ν and its strong Bochner integral satisfies the following norm inequality:(28)∫ETωdνωq≤∑σ∈Γλσ-2q-p∫ECωdνω, where q>p+1/2. In particular, ∫ET(ω)dν(ω)∈L.Proof.
We takeq>p+1/2. Then, by Lemma 11, for ν-a.e. ω∈E, we have(29)Tωξ-q≤Cω∑σ∈Γλσ-2q-pξq,σ∈Γ,which means that T(·) is an operator from Sq(M) to Sq∗(M). On the other hand, according to (1), for any ξ,η∈Sq∗(M), 〈〈T(ω)ξ,η〉〉:E→C is measurable. And Sq∗(M) is a separable space, so T(ω)ξ:E→Sq∗(M) is strongly measurable for any ξ∈Sq∗(M), where, together with (27), for any ξ∈Sq∗(M), the map(30)T·ξ:E⟶Sq∗Mis Bochner-integrable with respect to ν. Thus, T(·):E→L is strongly Bochner-integrable with respect to ν. Finally, (28) easily follows from (29) and we find ∫ET(ω)dν(ω)∈L.Definition 17.
Let0≤L<+∞; the map T(·):[0,L]→L is said to be sequentially continuous; if (tn)n≥1 is a sequence in [0,L] and (tn)n≥1 converges to t0∈[0,L], we have that T(tn) converges to T(t0).Definition17 is the notion of sequentially continuous, because L-valued map T(·) is defined on interval, so it is equivalent to continuous. The next theorem offers a necessary and sufficient condition for L-valued functions to be continuous.Theorem 18.
Let0≤L<+∞; T(·):[0,L]→L is a L-valued function; then T(·) is continuous if and only if(1)
T·^(σ,τ):[0,L]→C is continuous for any σ,τ∈Γ;(2)
there existp≥0 and C≥0 such that(31)T·^σ,τ≤Cλσpλτpfor eachσ,τ∈Γ and t∈[0,L].Proof.
We just prove the “if” part. Lett0∈[0,L] and (tn)n≥1⊂[0,L] converges to t0; then the sequence (T(tn))n≥1∈L and satisfies the following conditions:
(1) For everyσ∈Γ, T(tn)^(σ,τ)→T(t0)^(σ,τ).
(2) There existp≥0 and C≥0, such that(32)Ttn^σ,τ≤Cλσpλτp,σ∈Γ,which implies that the operator sequence (T(tn))n≥1 satisfies the conditions in Theorem 14; thus, there exists S∈L such that T(tn)→S, and(33)Tt0^σ,τ=limn→∞Ttn^σ,τ=S^σ,τ.Then T(t0)=S, which implies that(34)Ttn⟶Tt0,and T(·) is continuous.


## 4. Application to Quantum SDEs

In the present section, we show applications of our main results in Section3. We establish the existence and uniqueness of solution to (1). Equation (1) can describe quantum stochastic evolutions in the presence of quantum noise. As usual, by a solution of (1) we mean a map T:R+→L which satisfies the following integral equation:(35)Tt=J+∫0tXs,Tsds+∫0tYs,Ts∗Nsds,t∈R+,where the integral is the Bochner integral of operator function in L.Definition 19.
LetΘ:R+×L→L be a map. Θ is said to satisfy a locally uniform Lipschitz condition and a locally uniform linear growth condition, if, for any L≥0, there exists a constant ρ≥0, such that, for any σ,τ∈Γ and A,B∈L, one has(36)Θt,A^σ,τ-Θt,B^σ,τ≤ρA^σ,τ-B^σ,τ,t∈0,L,Θt,A^σ,τ≤ρ1+A^σ,τ,t∈0,L.Theorem 20.
LetX,Y:R+×L→L be continuous and satisfy a locally uniform Lipschitz condition and locally uniform linear growth condition, and let the noise process (N(t))t∈R+ satisfy the following:(1)
N:R+→L is continuous.(2)
For anyL≥0, there exists a constant K≥1, such that, for any σ,τ∈Γ,(37)Nt^σ,τ≤K,t∈0,L.Then, for any J∈L, there exists a unique continuous map T:R+→L, solving (1).Proof.
We just prove that, for anyL≥0, the integral equation (35) has a unique continuous solution on [0,L].
First, using iteration, we will prove that, for eachn≥1, there exists a sequence Tn:[0,L]→L, which has the following properties:
(1) For anyσ,τ∈Γ, the function Tn(·)^(σ,τ):[0,L]→C is continuous.
(2) For anyσ,τ∈Γ and t∈[0,L], there exists a constant α>0, such that(38)Tnt^σ,τ≤αλσpλτp+1+αλσpλτp∑i=1nρitiKii!.
(3)(39)Tnt=J+∫0tXs,Tn-1sds+∫0tYs,Tn-1s∗Nsds,where T0(s)=J.
In fact, from the assumptions and Theorem16, for any t∈[0,L], the integral ∫0tX(s,J)ds and ∫0tY(s,J)∗N(s)ds exist and belong to L. Now we define T1:[0,L]→L as follows:(40)T1t=J+∫0tXs,Jds+∫0tYs,J∗Nsds,t∈0,L.It is easy to see that, for any σ,τ∈Γ, the function T1(·)^(σ,τ):[0,L]→C is continuous. On the other hand, from the assumptions, there exist constants p≥0, α>0, and ρ>0, such that(41)T1t^σ,τ≤J^σ,τ+∫0tXs,J^σ,τds+∫0tYs,J^σ,τNs^σ,τds≤αλσpλτp+K∫0tXs,J^σ,τ+Ys,J^σ,τds≤αλσpλτp+Kρ1+J^σ,τt≤αλσpλτp+1+αλσpλτp∑i=11ρitiKii!,where σ,τ∈Γ and t∈[0,L].
Then we assume that, for somen≥1 and t∈[0,L], Tn(t) has been defined and has the properties shown above. Then, by Theorem 18, we find that Tn(·) is continuous; thus(42)Y·,Tn·:0,L⟶Cis also continuous, which means that the function Y(s,Tn(s))^(σ,τ)N(s)^(σ,τ), s∈[0,L], is measurable. On the other hand, we have(43)Ys,Tns^σ,τNs^σ,τ≤α1λσpλτp,σ,τ,∈Γ,s∈0,L,where α1=Kρ(1+eρLK+α+αeρLK). By Theorem 16, for any t∈[0,L], the integral ∫0tY(s,Tn(s))∗N(s)ds exists and belongs to L. Similarly, for any t∈[0,L], the integral ∫0tX(s,Tn(s))ds also exists and belongs to L.
Now we define a mapTn+1:[0,L]→L as follows:(44)Tn+1t=J+∫0tXs,Tnsds+∫0tYs,Tns∗Nsds,t∈0,L.Obviously, for any σ,τ∈Γ, the function Tn+1(·)^(σ,τ):[0,L]→C is continuous. Furthermore, because X and Y satisfy a locally uniform Lipschitz condition and locally uniform linear growth condition, we have(45)Tn+1t^σ,τ≤J^σ,τ+∫0tXs,Tns^σ,τds+∫0tYs,Tns^σ,τNs^σ,τds≤αλσpλτp+K∫0tXs,Tns^σ,τ+Ys,Tns^σ,τds≤αλσpλτp+Kρ∫0t1+Tns^σ,τds≤αλσpλτp+Kρt+Kρ∫0tTns^σ,τds.Note that(46)∫0tTns^σ,τds≤αtλσpλτp+1+αλσpλτp∑i=1nρiKiti+1i+1!.So we obtain that, for any σ,τ∈Γ,(47)∫0tTn+1s^σ,τds≤αλσpλτp+Kρt+Kραtλσpλτp+1+αλσpλτp∑i=1nρiKiti+1i+1!≤αλσpλτp+1+αλσpλτp∑i=1n+1ρiKitii!.Therefore, by induction, we come to our result.
Next, we show that, for anyt∈[0,L], (Tn(t))n≥1 converges in L. Indeed, by induction, for any σ,τ∈Γ and t∈[0,L], we have(48)Tnt^σ,τ-Tn-1t^σ,τ≤1+αλσpλτpρnKntnn!.So, for any t∈[0,L] and σ,τ∈Γ, the sequence (Tn(t)^(σ,τ))n≥1 converges in L. On the other hand, from the properties of Tn(t), we have(49)Tnt^σ,τ≤α2λσpλτp,where α2=α+eρKt+αeρKt. Thus, it follows from Theorem 14 that for any t∈[0,L], (Tn(t))n≥1 converges in L.
We let(50)Tt=limn→∞Tnt,t∈0,L.Then, it is easy to see that(51)Tt^σ,τ≤α2λσpλτp.Moreover, for t∈[0,L], n,m≥1, and σ,τ∈Γ, we have(52)Tn+mt^σ,τ-Tnt^σ,τ≤∑i=n+1n+mTit^σ,τ-Ti-1t^σ,τ≤1+αλσpλτp∑i=n+1∞ρiKitii!.So(53)sup0≤t≤LTt^σ,τ-Tnt^σ,τ≤1+αλσpλτp∑i=n+1∞ρiKiLii!,which means that the function T(·)^(σ,τ):[0,L]→C is continuous. Then, by Theorem 18, the function T(·):[0,L]→L is continuous.
In the same manner as above, we can show that, for eacht∈[0,L], the integrals ∫0tX(s,T(s))ds and ∫0tY(s,T(s))∗N(s)ds exist and belong to L. Further, by the dominated convergence theorem, we find that, for any t∈[0,L] and σ,τ∈Γ,(54)∫0tXs,Tns^σ,τds⟶∫0tXs,Ts^σ,τds,∫0tYs,Tns^σ,τNs^σ,τds⟶∫0tYs,Ts^σ,τNs^σ,τds.Note that, for any t∈[0,L] and σ,τ∈Γ, we have(55)Tn+1t^σ,τ=J^σ,τ+∫0tXs,Tns^σ,τds+∫0tYs,Tns^σ,τNs^σ,τds.Thus, by taking limit, we get(56)Tt^σ,τ=J^σ,τ+∫0tXs,Ts^σ,τds+∫0tYs,Ts^σ,τNs^σ,τds.which means that, for any t∈[0,L], we have(57)Tt=J+∫0tXs,Tsds+∫0tYs,Ts∗Nsds.So the map T:[0,L]→L is a continuous solution to (35).
Finally, we prove the uniqueness. Assume thatS:[0,L]→L is another continuous solution to (35). Then together with X and Y satisfying a locally uniform Lipschitz condition and locally uniform linear growth condition, for each t∈[0,L] and σ,τ∈Γ, we have(58)Tt^σ,τ-St^σ,τ≤∫0tXs,Ts^σ,τ-Xs,Ss^σ,τds+∫0tYs,Ts^σ,τ-Ys,Ss^σ,τNs^σ,τds≤ρK∫0tTs^σ,τ-Ss^σ,τds.By Gronwall inequality, we obtain that (59)Tt^σ,τ-St^σ,τ=0,t∈0,L.By Lemma 10, T(t)=S(t), t∈[0,L]; that is, the solution is unique.In the following, we always assume that the mapX,Y:R+×L→L and the noise process (N(t))t∈R+ are the same as in Theorem 20. For J∈L, we denote by {T(t,J)∣t∈R+} the solutions of (1) with J being its initial value. The following theorem shows that the solution to (1) continuously depends on the initial value.Theorem 21.
LetJ1, J2∈L be given. Then, for any 0≤L<+∞, there exists constant ρ≥0, such that, for any σ,τ∈Γ and t,s∈[0,L], one has(60)Tt,J1^σ,τ-Ts,J2^σ,τ≤eρKLJ1^σ,τ-J2^σ,τ+t-s1+J1^σ,τ+J2^σ,τ.Proof.
For0≤L<+∞, let p≥0, ρ>0, and K≥1 be such that, for all t∈[0,L] and σ,τ∈Γ, we have(1)
Nt^σ,τ≤K;(2)
Xt,A^σ,τ-Xt,B^σ,τ+Yt,A^σ,τ-Yt,B^σ,τ≤ρA^σ,τ-B^σ,τ, A,B∈L;(3)
Xt,A^σ,τ+Xt,B^σ,τ≤ρ1+A^σ,τ.
It is easy to see that, for anyA∈L,(61)Tt,A^σ,τ≤A^σ,τ+ρK∫0t1+Ts,A^σ,τds,which implies that(62)1+Tt,A^σ,τ≤1+A^σ,τeρKL. Similarly, we can prove that(63)Tt,J1^σ,τ-Tt,J2^σ,τ≤J1^σ,τ-J2^σ,τeρKL. Then, using estimates (62) and (63), we have(64)Tt,J1^σ,τ-Ts,J2^σ,τ≤Tt,J1^σ,τ-Tt,J2^σ,τ+Tt,J2^σ,τ-Ts,J2^σ,τ≤eρKLJ1^σ,τ-J2^σ,τ+t-s1+J1^σ,τ+J2^σ,τ.



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*Source: 8430975-2018-02-04.xml*