# Existence and Uniqueness of Solutions to Neutral Stochastic Functional Differential Equations with Poisson Jumps

Authors: Jianguo Tan; Hongli Wang; Yongfeng Guo
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

A class of neutral stochastic functional differential equations with Poisson jumps (NSFDEwPJs),d[x(t)-G(xt)]=f(xt,t)dt+g(xt,t)dW(t)+h(xt,t)dN(t),t∈[t0,T], with initial value xt0=ξ={ξ(θ):-τ≤θ≤0}, is investigated. First, we consider the existence and uniqueness of solutions to NSFDEwPJs under the uniform Lipschitz condition, the linear growth condition, and the contractive mapping. Then, the uniform Lipschitz condition is replaced by the local Lipschitz condition, and the existence and uniqueness theorem for NSFDEwPJs is also derived.


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## Body

## 1. Introduction

Neutral stochastic functional differential equations (NSFDEs) have recently been studied intensively, for instance, Mao [1] and Kolmanovskii [2], Mao et al. [3–6], Luo et al. [7], Zhou and Hu [8], and Luo [9]. Poisson jumps are becoming increasingly used to model real-world phenomena in different fields such as economics, finance, biology, and physics. There is an extensive literature concerned with Poisson jumps, for example, Wang et al. [10, 11], Ronghua et al. [12, 13], Luo [14], and Tan and Wang [15]. Therefore, it is natural and necessary to incorporate jumps in the neutral stochastic functional differential equations. However, the study of NSFDEwPJs is limited by far. Liu et al. [16] studied the stability of NSFDEwPJs by using fixed point theory, Luo and Taniguchi [17] proved the existence and uniqueness of non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jumps. However, no theory for the existence and uniqueness of solutions to NSFDEwPJs has been established yet. Therefore, in this paper, we first prove the existence and uniqueness of solutions to NSFDEwPJs.The outline of the paper is as follows. In Section2 we will introduce some necessary notations and assumptions. In Section 3, we will present several useful lemmas, and then we prove the existence and uniqueness of NSFDEwPJs under the uniform Lipschitz condition, the linear growth condition, and the contractive mapping. Furthermore, the uniform Lipschitz condition is replaced by the local Lipschitz condition, and the existence and uniqueness theorem is also derived.


## 2. Preliminaries

Let(Ω,ℱ,P) be a complete probability space with a filtration {ℱt}t≥0, which satisfies the usual conditions, that is, the filtration is continuous on the right and ℱ0 contains all P-null sets. Moreover, C([a,b];Rn) denotes the family of functions φ from [a,b] to Rn that are right-continuous and have limits on the left, C([a,b];Rn) is equipped with the norm ∥ψ∥=supa<t<b|ψ(t)|, where |·| is the Euclidean norm in Rn, that is, |x|=xTx(x∈Rn). Let ℒp([a,b];Rn) denote the family of all Rn-valued ℱt-adapted process f(t), t∈[a,b], such that ∫ab‍|f(t)|pdt<∞. We denote by ℳ2([-τ,0];Rn) the family of the processes ξ(t) in ℒ2([-τ,0];Rn) such that E∫-τ0‍|ξ(t)|2dt<∞. a∨b means the maximum of a and b, and a∧b means the minimum of a and b.In this paper, we consider then-dimensional neutral stochastic functional differential equations with Poisson jumps
(2.1)d[x(t)-G(xt)]=f(xt,t)dt+g(xt,t)dW(t)+h(xt,t)dN(t),t∈[t0,T],
where xt={x(t+θ):-τ≤θ≤0} can be regarded as a C([-τ,0];Rn)-valued stochastic process, W(t) is an m-dimensional standard Wiener process that is ℱt-adapted, and N(t) is a scalar Poisson process with intensity λ. Assume that W(t) and N(t) are independent of ℱt0. Moreover, the functions G:C([-τ,0];Rn)→Rn, f:C([-τ,0];Rn)×[t0,T]→Rn, g:C([-τ,0];Rn)×[t0,T]→Rn×m, h:C([-τ,0];Rn)×[t0,T]→Rn.The stochastic integral is defined in the Itô sense, and the integral version of (2.1) is frequently expressed as
(2.2)x(t)-G(xt)=x(t0)-G(xt0)+∫t0t‍f(xs,s)ds+∫t0t‍g(xs,s)dW(s)+∫t0t‍h(xs,s)dN(s).The initial value is followed:(2.3)xt0=ξ={ξ(θ):-τ≤θ≤0}isFt0-measurableC([-τ,0];Rn)-valuedrandomvariablesuchthatξ∈M2([-τ,0];Rn).


## 3. The Existence and Uniqueness Theorem

Definition 3.1.
AnRn-valued stochastic process x(t) on t0-τ≤t≤T is called a solution to (2.1) with initial data (2.3) if it has the following properties:(i)
x(t) is continuous and {xt}t0≤t≤T is ℱt-adapted;(ii)
f(xt,t)∈ℒ1([t0,T];Rn) and g(xt,t)∈ℒ2([t0,T];Rn×m);(iii)
xt0=ξ and (2.2) hold for each t0≤t≤T.A solutionx(t) is said to be unique if any other solution x-(t) is indistinguishable from it, that is,
(3.1)P{x(t)=x-(t)∀t0-τ≤t≤T}=1.Let us now begin to establish the theory of the existence and uniqueness of the solution to (2.1) with initial data (2.3).Theorem 3.2.
Assume that there exist two positive constantsK- and K such that(H1)
(the Lipschitz condition) For allϕ,φ∈C([-τ,0];Rn) and t∈[t0,T],
(3.2)|f(ϕ,t)-f(φ,t)|2∨|g(ϕ,t)-g(φ,t)|2∨|h(ϕ,t)-h(φ,t)|2≤K-‖ϕ-φ‖2.(H2)
(the linear growth condition) for all(ϕ,t)∈C([-τ,0];Rn)×[t0,T],
(3.3)|f(ϕ,t)|2∨|g(ϕ,t)|2∨|h(ϕ,t)|2≤K(1+‖ϕ‖2).(H3)
(the constractive mapping) there is a positive constantκ∈(0,1) such that, for all ϕ, φ∈C([-τ,0];Rn),
(3.4)|G(ϕ)-G(φ)|≤κ‖ϕ-φ‖.
Then, there exists a unique solutionx(t) to (2.1) with initial data (2.3). Moreover, the solution belongs to ℳ2([t0-τ,T];Rn).To prove Theorem3.2, we give the following useful Lemmas 3.3 and 3.4 first.Lemma 3.3.
Ifp≥2, g∈ℳ2([t0,T];Rn×m) such that
(3.5)E∫t0t‍|g(s)|pds<∞,
then E|∫t0T‍g(s)dW(s)|p≤(p(p-1)/2)p/2T(p-2)/2E∫t0T‍|g(s)|pds.
In particular, forp=2, there is equality.The proof of Lemma3.3 can be found in [6].Lemma 3.4.
For anya, b>0 and 0<α<1 one has
(3.6)(a+b)2≤a2α+b21-α.The proof of Lemma3.4 can also be found in [6].To prove the uniqueness of the solution for (2.1), we will establish Lemma 3.5.Lemma 3.5.
Let (3.3) and (3.4) hold. If x(t) is a solution to (2.1) with initial data (2.3), then
(3.7)E(sup⁡t0-τ≤t≤T|x(t)|2)≤(1+5+κκ(1-κ)(1-κ)E‖ξ‖2)×exp⁡[4K(T-t0)(T-t0+4K+8λ+2λ2(T-t0))(1-κ)(1-κ)],
where κ is the contraction constant in (3.4). In particular, x(t) belongs to ℳ2([t0-τ,T];Rn).Proof.
For every integern≥1, define the stopping time
(3.8)τn=T∧inf⁡{t∈[t0,T]:‖xt‖≥n}.
Clearly, τn↑T a.s. Set xn(t)=x(t∧τn) for t∈[t0,T]. Then, for t0≤t≤T(3.9)xn(t)=G(xtn)-G(ξ)+ξ(0)+∫t0t‍f(xsn,s)I[t0,τn](s)ds+∫t0t‍g(xsn,s)I[t0,τn](s)dW(s)+∫t0t‍h(xsn,s)I[t0,τn](s)dN(s),
where IA is the indicator function of a set A, that is,
(3.10)IA(x)={1,x∈A,0,x∉A.
Set
(3.11)Jn(t)=ξ(0)+∫t0t‍f(xsn,s)I[t0,τn]ds+∫t0t‍g(xsn,s)I[t0,τn]dW(s)+∫t0t‍h(xsn,s)I[t0,τn]dN(s).
Appling Lemma 3.4 twice and (3.4), one derives that
(3.12)|xn(t)|2≤1κ|G(xtn)-G(ξ)|2+11-κ|Jn(t)|2≤1κκ2‖xtn-ξ‖2+11-κ|Jn(t)|2=κ‖xtn-ξ‖2+11-κ|Jn(t)|2≤κ(1κ‖xtn‖2+11-κ‖ξ‖2)+11-κ|Jn(t)|2=κ‖xtn‖2+κ1-κ‖ξ‖2+11-κ|Jn(t)|2.
Hence,
(3.13)E(sup⁡t0≤s≤t|xn(s)|2)≤κE(sup⁡t0-τ≤s≤t|xn(s)|2)+κ1-κE‖ξ‖2+11-κE(sup⁡t0≤s≤t|Jn(s)|2).
Noting that E(supt0-τ≤s≤t|xn(s)|2)≤E∥ξ∥2+E(supt0≤s≤t|xn(s)|2), one sees that
(3.14)E(sup⁡t0-τ≤s≤t|xn(s)|2)≤κE(sup⁡t0-τ≤s≤t|xn(s)|2)+1+κ-κ1-κE‖ξ‖2+11-κE(sup⁡t0≤s≤t|Jn(s)|2).
Consequently,
(3.15)E(sup⁡t0-τ≤s≤t|xn(s)|2)≤1+κ-κ(1-κ)2E‖ξ‖2+1(1-κ)(1-κ)E(sup⁡t0≤s≤t|Jn(s)|2).
Using the basic inequality |a+b+c+d|2≤4|a|2+4|b|2+4|c|2+4|d|2 and the Hölder inequality, and for the jump integral, we can transform to the compensated Poisson process N~(t)=N(t)-λt, and we can get
(3.16)|Jn(t)|2≤4|ξ(0)|2+4|∫t0t‍f(xsn,s)I[t0,τn](s)ds|2+4|∫t0t‍g(xsn,s)I[t0,τn](s)dW(s)|2+4|∫t0t‍h(xsn,s)I[t0,τn](s)dN(s)|2≤4|ξ(0)|2+4(t-t0)∫t0t‍|f(xsn,s)|2ds+4|∫t0t‍g(xsn,s)I[t0,τn](s)dW(s)|2+4|∫t0t‍h(xsn,s)I[t0,τn](s)dN~(s)+λ∫t0t‍h(xsn,s)I[t0,τn](s)ds|2≤4|ξ(0)|2+4(t-t0)∫t0t‍|f(xsn,s)|2ds+4|∫t0t‍g(xsn,s)I[t0,τn](s)dW(s)|2+8|∫t0t‍h(xsn,s)I[t0,τn](s)dN~(s)|2+8λ2(t-t0)∫t0t‍|h(xsn,s)|2ds.
Since both the Wiener process W(t) and compensated Poisson process N~(t) are martingales, using the Doob martingale inequality, we have
(3.17)E(sup⁡t0≤s≤t|Jn(s)|2)≤4E|ξ(0)|2+4(t-t0)E∫t0t‍|f(xsn,s)|2ds+4×4E∫t0t‍|g(xsn,s)|2ds+4(2×4λ+2λ2(t-t0))E∫t0t‍|h(xsn,s)|2ds≤4E‖ξ‖2+4K(T-t0+4+8λ+2λ2(T-t0))∫t0t‍(1+E‖xsn‖2)ds.
Substituting this into (3.15),
(3.18)E(sup⁡t0-τ≤s≤t|xn(s)|2)≤5+κκ(1-κ)(1-κ)E‖ξ‖2+4K(T-t0+4+8λ+2λ2(T-t0))(1-κ)(1-κ)∫t0t‍(1+E‖xsn‖2)ds.
Therefore,
(3.19)1+E(sup⁡t0-τ≤s≤t|xn(s)|2)≤1+5+κκ(1-κ)(1-κ)E‖ξ‖2+4K(T-t0+4+8λ+2λ2(T-t0))(1-κ)(1-κ)∫t0t‍(1+E(sup⁡t0-τ≤r≤s|xn(r)|2))ds.
Now the Gronwall inequality yields that
(3.20)1+E(sup⁡t0-τ≤t≤T|xn(t)|2)≤(1+5+κκ(1-κ)(1-κ)E‖ξ‖2)×exp⁡[4K(T-t0)(T-t0+4+8λ+2λ2(T-t0))(1-κ)(1-κ)].
Consequently
(3.21)E(sup⁡t0-τ≤t≤τn|x(t)|2)≤(1+5+κκ(1-κ)(1-κ)E‖ξ‖2)×exp⁡[4K(T-t0)(T-t0+4+8λ+2λ2(T-t0))(1-κ)(1-κ)].
Letting n→∞, it then implies the following inequality:
(3.22)E(sup⁡t0-τ≤t≤T|x(t)|2)≤(1+5+κκ(1-κ)(1-κ)E‖ξ‖2)×exp⁡[4K(T-t0)(T-t0+4+8λ+2λ2(T-t0))(1-κ)(1-κ)].
The proof is complete.Proof of Theorem3.2.
Uniqueness. Let x(t) and x-(t) be two solutions to (2.1). By Lemma 3.5 both of them belong to ℳ2([t0-τ,T];Rn). Note that
(3.23)x(t)-x-(t)=G(xt)-G(x-t)+J(t),
where
(3.24)J(t)=∫t0t‍[f(xs,s)-f(x-s,s)]ds+∫t0t‍[g(xs,s)-g(x-s,s)]dW(s)+∫t0t‍[h(xs,s)-h(x-s,s)]dN(s).
By Lemma 3.4 and condition (3.4), one sees easily that
(3.25)|x(t)-x-(t)|2≤1κ|G(xt)-G(x-t)|2+11-κ|J(t)|2≤κ‖xt-x-t‖2+11-κ|J(t)|2.
Therefore,
(3.26)Esup⁡t0≤s≤t|x(s)-x-(s)|2≤κEsup⁡t0≤s≤t|x(s)-x-(s)|2+11-κEsup⁡t0≤s≤t|J(s)|2,
which implies that
(3.27)Esup⁡t0≤s≤t|x(s)-x-(s)|2≤1(1-κ)2Esup⁡t0≤s≤t|J(s)|2.
On the other hand,
(3.28)Esup⁡t0≤s≤t|J(s)|2≤3K-(T-t0+4+8λ+2λ2(T-t0))∫t0t‍E‖xs-x-s‖2ds=3K-(T-t0+4+8λ+2λ2(T-t0))∫t0t‍Esup⁡t0-τ≤r≤s|x(r)-x-(r)|2ds.
Noting the fact that x(r)=x-(r)=ξ, for r∈[t0-τ,t0], we have
(3.29)sup⁡t0-τ≤r≤s|x(r)-x-(r)|2=sup⁡t0≤r≤s|x(r)-x-(r)|2.
Then
(3.30)Esup⁡t0≤s≤t|J(s)|2≤3K-(T-t0+4+8λ+2λ2(T-t0))∫t0t‍Esup⁡t0≤r≤s|x(r)-x-(r)|2ds.
Therefore,
(3.31)Esup⁡t0≤s≤t|x(s)-x-(s)|2≤3K-(T-t0+4+8λ+2λ2(T-t0))(1-κ)2∫t0t‍Esup⁡t0≤r≤s|x(r)-x-(r)|2ds.
The Grownwall inequality then yields that
(3.32)Esup⁡t0≤t≤T|x(t)-x-(t)|2=0.
Therefore the uniqueness has been proved.
Existence. We divide the whole proof of the existence into two steps.
Step  1. We impose an additional condition: T-t0 is sufficiently small so that
(3.33)δ:=κ+3K-(T-t0+4+8λ+2λ2(T-t0))1-κ(T-t0)<1.
Define xt00=ξ and x0(t)=ξ(0) for t0≤t≤T. For each n=1,2,…, set xt0n=ξ and define, by the Picard iterations
(3.34)xn(t)-G(xtn-1)=ξ(0)-G(ξ)+∫t0t‍f(xsn-1,s)ds+∫t0t‍g(xsn-1,s)dW(s)+∫t0t‍h(xsn-1,s)dN(s).
Obviously, x0(t)∈ℳ2([t0-τ,T];Rn), and by induction xn(t)∈ℳ2([t0-τ,T];Rn). In fact,
(3.35)xn(t)=G(xtn-1)-G(ξ)+ξ(0)+∫t0t‍f(xsn-1,s)ds+∫t0t‍g(xsn-1,s)dW(s)+∫t0t‍h(xsn-1,s)dN(s).
Let
(3.36)Pn-1(t)=ξ(0)+∫t0t‍f(xsn-1,s)ds+∫t0t‍g(xsn-1,s)dW(s)+∫t0t‍h(xsn-1,s)dN(s).
Then by Lemma 3.4 and (3.4) we have
(3.37)|xn(t)|2≤1κ|G(xtn)-G(ξ)|2+11-κ|Pn-1(t)|2≤κ‖xtn-ξ‖2+11-κ|Pn-1(t)|2≤κ(1κ‖xtn‖2+11-κ‖ξ‖2)+11-κ|Pn-1(t)|2=κ‖xtn‖2+κ1-κ‖ξ‖2+11-κ|Pn-1(t)|2≤κ(‖ξ‖2+|xn(t)|2)+κ1-κ‖ξ‖2+11-κ|Pn-1(t)|2.
Then,
(3.38)|xn(t)|2≤κ(1-κ)2‖ξ‖2+1(1-κ)(1-κ)|Pn-1(t)|2.
Therefore,
(3.39)E|xn(t)|2≤κ(1-κ)2E‖ξ‖2+1(1-κ)(1-κ)E|Pn-1(t)|2.
By the basic inequality |a+b+c+d|2≤4|a|2+4|b|2+4|c|2+4|d|2, the Hölder inequality, Lemma 3.3, and martingale isometry, we have
(3.40)E|Pn-1(t)|2≤4E|ξ(0)|2+4E|∫t0t‍f(xsn-1,s)ds|2+4E|∫t0t‍g(xsn-1,s)dW(s)|2+4E|∫t0t‍h(xsn-1,s)dN(s)|2≤4E|ξ(0)|2+4(t-t0)E∫t0t‍|f(xsn-1,s)|2ds+4E∫t0t‍|g(xsn-1,s)|2ds+4(2λ+2λ2(t-t0))E∫t0t‍|h(xsn-1,s)|2ds≤4E‖ξ‖2+4K(T-t0+1)∫t0t‍(1+E‖xsn-1‖2)ds+4K(2λ+2λ2(T-t0))∫t0t‍(1+E‖xsn-1‖2)ds≤4E‖ξ‖2+4K(T-t0+1+2λ+2λ2(T-t0))(T-t0)+4K(T-t0+1+2λ+2λ2(T-t0))E∫t0t‍sup⁡t0≤r≤s|xn-1(r)|2ds≤4E‖ξ‖2+4K(T-t0+1+2λ+2λ2(T-t0))(T-t0)+4K(T-t0+1+2λ+2λ2(T-t0))∫t0t‍E|xn-1(s)|2ds.
Therefore,
(3.41)E|xn(t)|2≤κ(1-κ)2E‖ξ‖2+1(1-κ)(1-κ)E|Pn-1(t)|2≤κ(1-κ)2E‖ξ‖2+4(1-κ)(1-κ)E‖ξ‖2+4K(T-t0+1+2λ+2λ2(T-t0))(T-t0)(1-κ)(1-κ)+4K(T-t0+1+2λ+2λ2(T-t0))(1-κ)(1-κ)∫t0t‍E|xn-1(s)|2ds.
Hence, for any k≥1, we have
(3.42)max⁡1≤n≤kE|xn(t)|2≤[κ(1-κ)2+4(1-κ)(1-κ)]E‖ξ‖2+4K(T-t0+1+2λ+2λ2(T-t0))(T-t0)(1-κ)(1-κ)+4K(T-t0+1+2λ+2λ2(T-t0))(1-κ)(1-κ)∫t0t‍max⁡1≤n≤kE|xn-1(s)|2ds.
Note that
(3.43)max⁡1≤n≤kE|xn-1(t)|2=max⁡{E|ξ(0)|2,E|x1(s)|2,…,E|xk-1(s)|2}≤max⁡{E‖ξ‖2,E|x1(s)|2,…,E|xk-1(s)|2}=max⁡{E‖ξ‖2,max⁡1≤n≤kE|xn(t)|2}≤E‖ξ‖2+max⁡1≤n≤kE|xn(t)|2.
Therefore,
(3.44)max⁡1≤n≤kE|xn(t)|2≤[κ(1-κ)2+4(1-κ)(1-κ)]E‖ξ‖2+4K(T-t0+1+2λ+2λ2(T-t0))(T-t0)(1-κ)(1-κ)+4K(T-t0+1+2λ+2λ2(T-t0))(1-κ)(1-κ)∫t0t‍(E‖ξ‖2+max⁡1≤n≤kE|xn(t)|2)ds≤c1+c2∫t0t‍max⁡1≤n≤kE|xn(s)|2ds,
where c1=[(κ/(1-κ)2)+(4/(1-κ)(1-κ))+(4K(T-t0+1+2λ+2λ2(T-t0))/((1-κ)(1-κ)))E∥ξ∥2]E∥ξ∥2 + (4K(T-t0+1+2λ+2λ2(T-t0))(T-t0)/((1-κ)(1-κ))), c2=(4K(T-t0+1+2λ+2λ2(T-t0))/((1-κ)(1-κ))).
By the Gronwall inequality, we have(3.45)max⁡1≤n≤kE|xn(t)|2≤c1ec2(T-t0).
Because k is arbitrary, we have
(3.46)E|xn(t)|2≤c1ec2(T-t0).
Hence, for all n≥0, xn(t)∈ℳ2([t0-τ,T];Rn).
Note that, fort0≤t≤T,
(3.47)x1(t)-x0(t)=x1(t)-ξ(0)=G(xt0)-G(ξ)+∫t0t‍f(xs0,s)ds+∫t0t‍g(xs0,s)dW(s)+∫t0t‍h(xs0,s)dN(s).
In a similar way to that in the proof of the existence, we get that
(3.48)Esup⁡t0≤t≤T|x1(t)-x0(t)|2≤κEsup⁡t0≤t≤T‖xt0-ξ‖2+3K(T-t0+4+8λ+2λ2(T-t0))1-κ∫t0t‍(1+‖xt0‖)2ds≤2κE‖ξ‖2+3K(T-t0+4+8λ+2λ2(T-t0))1-κ(1+E‖ξ‖2)(T-t0):=C.
Note also that, for n≥1 and t0≤t≤T,
(3.49)xn+1(t)-xn(t)=G(xtn)-G(xtn-1)+∫t0t‍[f(xsn,s)-f(xsn-1,s)]ds+∫t0t‍[g(xsn,s)-g(xsn-1,s)]dW(s)+∫t0t‍[h(xsn,s)-h(xsn-1,s)]dN(s).
In the same way as in the proof of the uniqueness, one derives that
(3.50)Esup⁡t0≤t≤T|xn+1(t)-xn(t)|2≤κEsup⁡t0≤t≤T|xn(t)-xn-1(t)|2+3K-(T-t0+4+8λ+2λ2(T-t0))1-κ∫t0T‍Esup⁡t0≤s≤t|xn(s)-xn-1(s)|2dt≤δEsup⁡t0≤t≤T|xn(t)-xn-1(t)|2≤δnEsup⁡t0≤t≤T|x1(t)-x0(t)|2≤Cδn.
Next we verify that {xn(t)} converges to x(t) in the sense of L2 and probability 1 on ℳ2([t0-τ,T];Rn). Moreover, x(t) is the solution to (2.1) with initial data (2.3). Hence, by the Chebyshev inequality,
(3.51)P{sup⁡t0≤t≤T|xn+1(t)-xn(t)|2>12n}≤C(4δ)n.
Since ∑n=0∞C(4δ)n≤∞, the Borel-Cantelli lemma yields that, for almost all ω∈Ω, there exist a positive integer n0=n0(ω) such that
(3.52)sup⁡t0≤t≤T|xn+1(t)-xn(t)|2≤12n,n≥n0.
It follows that, with probability 1, the partial sum
(3.53)x0(t)+∑i=0n-1‍[xi+1(t)-xi(t)]=xn(t)
is the partial sum of function series
(3.54)x0(t)+|x1(t)-x0(t)|+⋯+|xn(t)-xn-1(t)|+⋯.
By the second item of series (3.54), the absolute value of every item (3.54) is less than the corresponding item of positive
(3.55)1+12+122+⋯+12n+⋯.
Moreover, the positive series is convergent; further, by the Weierstrass criterion, series (3.54) is convergent on [t0,T]. Furthermore, it is uniformly on [t0,T]. Let the sum function be x(t). Therefore, the approximate sequence {xn(t)} uniformly converges to x(t) on [t0,T] and is ℱt-adapted, hence x(t) is also continuous and ℱt-adapted. On the other hand, (3.50) implies that, for each t, sequence {xn(t)} is a Cauchy sequence in ℳ2([t0-τ,T];Rn) as well. Hence, we also have xn(t)→x(t) in ℳ2([t0-τ,T];Rn). Letting n→∞ in (3.46) gives
(3.56)E|x(t)|2≤c1ec2(T-t0),t0≤t≤T.
Therefore, by the use of the above result, we get that
(3.57)E∫t0-τT‍|x(s)|2ds=E∫t0-τt0‍|x(s)|2ds+E∫t0T‍|x(s)|2ds≤E∫t0-τt0‍|ξ(s)|2ds+E∫t0-τT‍c1ec2(T-t0)ds<∞.
That is, x(t)∈ℳ2([t0-τ,T];Rn).
Now we prove thatx(t) satisfies (2.1):
(3.58)E|∫t0t‍[f(xsn,s)-f(xs,s)]ds|2+E|∫t0t‍[g(xsn,s)-g(xs,s)]dW(s)2|+E|∫t0t‍[h(xsn,s)-h(xs,s)]dN(s)|2≤(t-t0)E∫t0t‍|f(xsn,s)-f(xs,s)|2ds+E∫t0t‍|g(xsn,s)-g(xs,s)|2ds+(2λ+2λ2(t-t0))E∫t0t‍|h(xsn,s)-h(xs,s)|2ds≤(t-t0)K-E∫t0t‍‖xsn-xs‖2ds+K-E∫t0t‍‖xsn-xs‖2ds+(2λ+2λ2(t-t0))K-E∫t0t‍‖xsn-xs‖ds≤K-(T-t0+1+2λ+2λ2(T-t0))E∫t0t‍‖xsn-xs‖ds≤K-(T-t0+1+2λ+2λ2(T-t0))∫t0t‍Esup⁡t0≤r≤s|xn(r)-x(r)|2ds≤K-(T-t0+1+2λ+2λ2(T-t0))∫t0T‍E|xn(s)-x(s)|2ds⟶0(n⟶∞).
Hence, we can let n→∞ in (3.34) to obtain that
(3.59)x(t)-G(xt)=ξ(0)-G(ξ)+∫t0t‍f(xs,s)ds+∫t0t‍g(xs,s)dW(s)+∫t0t‍h(xs,s)dN(s).
So x(t) is the solution to (2.1).
Step  2. We need to remove the additional condition (3.33). Let σ>0 be sufficiently small for
(3.60)δ:=κ+3K-(σ+4+8λ+2λ2σ)1-κσ<1.
By Step 1, there is a solution to (2.1) on [t0-τ,t0+σ]. Now consider (2.1) on [t0+σ,t0+2σ] with initial data xt0+σ. By Step 1 again, there is a solution to (2.1) on [t0+σ,t0+2σ]. Repeating this procedure we see that there is a solution to (2.1) on the entire interval [t0-τ,T]. The proof is complete.For NSFDEwPJs, we know that the global Lipschitz condition imposed on Theorem3.2 is a big restriction; now we will replace the global Lipschitz condition by the local Lipschitz condition. Then Theorem 3.6 follows.Theorem 3.6.
Assume that there exist two positive constantsKn and K such that(i)
(the local Lipschitz condition) for allϕ,φ∈C([-τ,0];Rn) with ∥ϕ∥∨|φ∥≤n and t∈[t0,T],
(3.61)|f(ϕ,t)-f(φ,t)|2∨|g(ϕ,t)-g(φ,t)|2∨|h(ϕ,t)-h(φ,t)|2≤Kn‖ϕ-φ‖2.(ii)
(the linear growth condition) for all(ϕ,t)∈C([-τ,0];Rn)×[t0,T],
(3.62)|f(ϕ,t)|2∨|g(ϕ,t)|2∨|h(ϕ,t)|2≤K(1+‖ϕ‖2).(iii)
(the constractive mapping) there is a positive constantκ∈(0,1) such that, for all ϕ,φ∈C([-τ,0];Rn),
(3.63)|G(ϕ)-G(φ)|≤κ‖ϕ-φ‖.Then there exists a unique solution x(t) to (2.1) with initial data (2.3). Moreover, the solution belongs to ℳ2([t0-τ,T];Rn).Proof.
For eachn≥1, define truncation functions fn,gn, and hn as follows:
(3.64)fn(xt,t)={f(xt,t)‖xt‖≤n,f(nxt‖xt‖,t)‖xt‖>n,gn(xt,t)={g(xt,t)‖xt‖≤n,g(nxt‖xn‖,t)‖xt‖>n,hn(xt,t)={h(xt,t)‖xt‖≤n,h(nxt‖xn‖,t)‖xt‖>n,
Then fn, gn, and hn satisfy conditions (3.2) and (3.3). By Theorem 3.2, there is a unique solution xn(t) and xn(t)∈ℳ2([t0-τ,T];Rn) to the equation
(3.65)xn(t)-G(xtn)=ξ(0)-G(ξ)+∫t0t‍fn(xsn,s)ds+∫t0t‍gn(xsn,s)dW(s)+∫t0t‍hn(xsn,s)dN(s),t∈[t0,T].
Certainly xn+1(t) is a unique solution to the equation
(3.66)xn+1(t)-G(xtn+1)=ξ(0)-G(ξ)+∫t0t‍fn+1(xsn+1,s)ds+∫t0t‍gn+1(xsn+1,s)dW(s)+∫t0t‍hn+1(xsn+1,s)dN(s),t∈[t0,T]
and xn+1(t)∈ℳ2([t0-τ,T];Rn) to the equation.
Now define the stopping time(3.67)τn=T∧inf⁡{t∈[t0,T]:‖xtn‖≥n}.
We can show that xn(t)=xn+1(t) for t∈[t0,T]. Then by Lemma 3.4, basic inequality, we have
(3.68)|xn+1(t)-xn(t)|2≤1κ|G(xtn+1)-G(xtn)|2+31-κ|∫t0t‍[fn+1(xsn+1,s)-fn(xsn,s)]ds|2+31-κ|∫t0t‍[gn+1(xsn+1,s)-gn(xsn,s)]dW(s)|2+31-κ|∫t0t‍[hn+1(xsn+1,s)-hn(xsn,s)]dN(s)|2.
Taking the expectation, by the Hölder inequality and Lemma3.3, we transform the Poisson process to the compensated Poisson process N~(t)=N(t)-λt, and, by martingale isometry, we have
(3.69)E|xn+1(t)-xn(t)|2≤κE‖xtn+1-xtn‖2+31-κ(t-t0)E∫t0t‍|fn+1(xsn+1,s)-fn(xsn,s)|2ds+31-κE∫t0t‍|gn+1(xsn+1,s)-gn(xsn,s)|2ds+31-κ(2λ+2λ2(t-t0))E∫t0t‍|hn+1(xsn+1,s)-hn(xsn,s)|2ds≤κE‖xtn+1-xtn‖2+61-κ(t-t0)E∫t0t‍|fn+1(xsn+1,s)-fn+1(xsn,s)|2ds+61-κ(t-t0)E∫t0t‍|fn+1(xsn,s)-fn(xsn,s)|2ds+61-κE∫t0t‍|gn+1(xsn+1,s)-gn+1(xsn,s)|2ds+61-κE∫t0t‍|gn+1(xsn,s)-gn(xsn,s)|2ds+61-κ(2λ+2λ2(t-t0))E∫t0t‍|hn+1(xsn+1,s)-hn+1(xsn,s)|2ds+61-κ(2λ+2λ2(t-t0))E∫t0t‍|hn+1(xsn,s)-hn(xsn,s)|2ds.
For t0<t<tn, we have know that
(3.70)fn+1(xsn,s)=fn(xsn,s)=f(xsn,s),gn+1(xsn,s)=gn(xsn,s)=g(xsn,s),hn+1(xsn,s)=hn(xsn,s)=h(xsn,s).
Again by xt0n+1=xt0n=ξ, we get that
(3.71)E|xn+1(s)-xn(s)|2≤κE‖xtn+1-xtn‖2+61-κ(t-t0)E∫t0t‍|fn+1(xsn+1,s)-fn+1(xsn,s)|2ds+61-κE∫t0t‍|gn+1(xsn+1,s)-gn+1(xsn,s)|2ds+61-κ(2λ+2λ2(t-t0))E∫t0t‍|hn+1(xsn+1,s)-hn+1(xsn,s)|2ds,=κEsup⁡t0-τ≤s≤t|xn+1(s)-xn(s)|2+61-κ(t-t0)E∫t0t‍|f(xsn+1,s)-f(xsn,s)|2ds+61-κE∫t0t‍|g(xsn+1,s)-g(xsn,s)|2ds+61-κ(2λ+2λ2(t-t0))E∫t0t‍|h(xsn+1,s)-h(xsn,s)|2ds.
Noting the fact that xn+1(s)=xn(s)=ξ, for s∈[t0-τ,t0], we get
(3.72)Esup⁡t0≤s≤t|xn+1(s)-xn(s)|2≤6[(t-t0)+1+2λ+2λ2(t-t0)](1-κ)2E∫t0t‍Kn‖xsn+1-xsn‖2ds≤6[(t-t0)+1+2λ+2λ2(t-t0)](1-κ)2KnE∫t0t‍sup⁡t0≤s≤t|xn+1(s)-xn(s)|2ds.
From the Gronwall inequality one sees that
(3.73)Esup⁡t0≤s≤t|xn+1(s)-xn(s)|2,t0<t<τn.
This means that, for t0<t<τn, we always have
(3.74)xn(t)=xn+1(t).
It is then deduced that τn is increasing, that is as n→∞, τn↑T a.s. By the linear growth condition, for all almost all ω∈Ω, there exists an integer n0=n0(ω) such that τn=T as n>n0. Now define x(t) by x(t)=xn0(t), t∈[t0,T]. Next to verify that x(t) is the solution to (2.1), by (3.74) x(t∧τn)=xn(t∧τn), it follows that
(3.75)x(t∧τn)-G(xt∧τnn)=ξ(0)-G(ξ)+∫t0t∧τn‍fn(xs,s)ds+∫t0t∧τn‍gn(xs,s)dW(s)+∫t0t∧τn‍hn(xs,s)dN(s),
that is,
(3.76)x(t∧τn)-G(xt∧τnn)=ξ(0)-G(ξ)+∫t0t∧τn‍f(xs,s)ds+∫t0t∧τn‍g(xs,s)dW(s)+∫t0t∧τn‍h(xs,s)dN(s).
Letting n→∞(3.77)x(t∧T)-G(xtn∧T)=ξ(0)-G(ξ)+∫t0t∧T‍f(xs,s)ds+∫t0t∧T‍g(xs,s)dW(s)+∫t0t∧T‍h(xs,s)dN(s),
that is
(3.78)x(t)-G(xt)=ξ(0)-G(ξ)+∫t0t‍f(xs,s)ds+∫t0t‍g(xs,s)dW(s)+∫t0t‍h(xs,s)dN(s).
We can see that x(t) is the solution to (2.1) and x(t)∈ℳ2([t0-τ,T];Rn). The proof of existence is complete. By stopping our process, uniqueness is obtained. This completes the proof.



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*Source: 371239-2012-07-08.xml*