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

Abstract

A class of neutral stochastic functional differential equations with Poisson jumps (NSFDEwPJs), d[𝑥(𝑡)−𝐺(𝑥_𝑡)]=𝑓(𝑥_𝑡,𝑡)d𝑡+𝑔(𝑥_𝑡,𝑡)d𝑊(𝑡)+ℎ(𝑥_𝑡,𝑡)d𝑁(𝑡),𝑡∈[𝑡₀,𝑇], with initial value 𝑥_𝑡0=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.

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 Section 2 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 {ℱ_𝑡}_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];𝑅^𝑛) denotes the family of functions 𝜑 from [𝑎,𝑏] to 𝑅^𝑛 that are right-continuous and have limits on the left, 𝐶([𝑎,𝑏];𝑅^𝑛) is equipped with the norm ‖𝜓‖=sup_{𝑎<𝑡<𝑏}|𝜓(𝑡)|, where |⋅| is the Euclidean norm in 𝑅^𝑛, that is, |𝑥|=𝑥^𝑇𝑥(𝑥∈𝑅^𝑛). Let ℒ^𝑝([𝑎,𝑏];𝑅^𝑛) denote the family of all 𝑅^𝑛-valued ℱ_𝑡-adapted process 𝑓(𝑡), 𝑡∈[𝑎,𝑏], such that ∫^𝑏_𝑎|𝑓(𝑡)|^𝑝d𝑡<∞. We denote by ℳ²([−𝜏,0];𝑅^𝑛) the family of the processes 𝜉(𝑡) in ℒ²([−𝜏,0];𝑅^𝑛) such that 𝐸∫⁰_−𝜏|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 the 𝑛-dimensional neutral stochastic functional differential equations with Poisson jumps d𝑥(𝑡)−𝐺𝑥_𝑡=𝑓𝑥_𝑡,𝑡d𝑡+𝑔𝑥_𝑡,𝑡d𝑊(𝑡)+ℎ𝑥_𝑡,𝑡d𝑁(𝑡),𝑡∈𝑡₀,𝑇, (2.1) where 𝑥_𝑡={𝑥(𝑡+𝜃)∶−𝜏≤𝜃≤0} can be regarded as a 𝐶([−𝜏,0];𝑅^𝑛)-valued stochastic process, 𝑊(𝑡) is an 𝑚-dimensional standard Wiener process that is ℱ_𝑡-adapted, and 𝑁(𝑡) is a scalar Poisson process with intensity 𝜆. Assume that 𝑊(𝑡) and 𝑁(𝑡) are independent of ℱ_𝑡0. Moreover, the functions 𝐺∶𝐶([−𝜏,0];𝑅^𝑛)→𝑅^𝑛, 𝑓∶𝐶([−𝜏,0];𝑅^𝑛)×[𝑡₀,𝑇]→𝑅^𝑛, 𝑔∶𝐶([−𝜏,0];𝑅^𝑛)×[𝑡₀,𝑇]→𝑅^𝑛×𝑚, ℎ∶𝐶([−𝜏,0];𝑅^𝑛)×[𝑡₀,𝑇]→𝑅^𝑛.

The stochastic integral is defined in the Itô sense, and the integral version of (2.1) is frequently expressed as 𝑥(𝑡)−𝐺𝑥_𝑡=𝑥𝑡₀−𝐺𝑥_𝑡0+^𝑡_𝑡0𝑓𝑥_𝑠,𝑠d𝑠+^𝑡_𝑡0𝑔𝑥_𝑠,𝑠d𝑊(𝑠)+^𝑡_𝑡0ℎ𝑥_𝑠,𝑠d𝑁(𝑠). (2.2)

The initial value is followed: 𝑥_𝑡0=𝜉={𝜉(𝜃)∶−𝜏≤𝜃≤0}isℱ_𝑡0−measurable𝐶[−𝜏,0];𝑅^𝑛−valuedrandomvariablesuchthat𝜉∈ℳ²[−𝜏,0];𝑅^𝑛. (2.3)

3. The Existence and Uniqueness Theorem

Definition 3.1. An 𝑅^𝑛-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 {𝑥_𝑡}_{𝑡0≤𝑡≤𝑇} is ℱ_𝑡-adapted;(ii)𝑓(𝑥_𝑡,𝑡)∈ℒ¹([𝑡₀,𝑇];𝑅^𝑛) and 𝑔(𝑥_𝑡,𝑡)∈ℒ²([𝑡₀,𝑇];𝑅^𝑛×𝑚);(iii)𝑥_𝑡0=𝜉 and (2.2) hold for each 𝑡₀≤𝑡≤𝑇.

A solution 𝑥(𝑡) is said to be unique if any other solution 𝑥(𝑡) is indistinguishable from it, that is, 𝑃𝑥(𝑡)=𝑥(𝑡)∀𝑡₀−𝜏≤𝑡≤𝑇=1. (3.1)

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 constants 𝐾 and 𝐾 such that(H1) (the Lipschitz condition) For all 𝜙,𝜑∈𝐶([−𝜏,0];𝑅^𝑛) and 𝑡∈[𝑡₀,𝑇], 𝑓𝜙,𝑡−𝑓𝜑,𝑡²∨𝑔𝜙,𝑡−𝑔𝜑,𝑡²∨ℎ𝜙,𝑡−ℎ𝜑,𝑡²≤𝐾𝜙−𝜑². (3.2)(H2) (the linear growth condition) for all (𝜙,𝑡)∈𝐶([−𝜏,0];𝑅^𝑛)×[𝑡₀,𝑇], 𝑓𝜙,𝑡²∨𝑔𝜙,𝑡²∨ℎ𝜙,𝑡²≤𝐾1+𝜙². (3.3)(H3) (the constractive mapping) there is a positive constant 𝜅∈(0,1) such that, for all 𝜙, 𝜑∈𝐶([−𝜏,0];𝑅^𝑛), 𝐺𝜙−𝐺𝜑≤𝜅𝜙−𝜑. (3.4)
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];𝑅^𝑛).

To prove Theorem 3.2, we give the following useful Lemmas 3.3 and 3.4 first.

Lemma 3.3. If 𝑝≥2, 𝑔∈ℳ²([𝑡₀,𝑇];𝑅^𝑛×𝑚) such that 𝐸^𝑡_𝑡0𝑔(𝑠)^𝑝d𝑠<∞, (3.5) then 𝐸|∫^𝑇_𝑡0𝑔(𝑠)d𝑊(𝑠)|^𝑝≤(𝑝(𝑝−1)/2)^𝑝/2𝑇^(𝑝−2)/2𝐸∫^𝑇_𝑡0|𝑔(𝑠)|^𝑝d𝑠.
In particular, for 𝑝=2, there is equality.

The proof of Lemma 3.3 can be found in [6].

Lemma 3.4. For any 𝑎, 𝑏>0 and 0<𝛼<1 one has (𝑎+𝑏)²≤𝑎²𝛼+𝑏².1−𝛼 (3.6)

The proof of Lemma 3.4 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 𝐸sup_{𝑡0−𝜏≤𝑡≤𝑇}|𝑥(𝑡)|²≤√1+5+𝜅𝜅(1−𝜅)1−√𝜅𝐸𝜉²×exp4𝐾𝑇−𝑡₀𝑇−𝑡₀+4𝐾+8𝜆+2𝜆²𝑇−𝑡₀(1−𝜅)1−√𝜅, (3.7) where 𝜅 is the contraction constant in (3.4). In particular, 𝑥(𝑡) belongs to ℳ²([𝑡₀−𝜏,𝑇];𝑅^𝑛).

Proof. For every integer 𝑛≥1, define the stopping time 𝜏_𝑛=𝑇∧inf𝑡∈𝑡₀,𝑇∶𝑥_𝑡≥𝑛. (3.8) Clearly, 𝜏_𝑛↑𝑇 a.s. Set 𝑥^𝑛(𝑡)=𝑥(𝑡∧𝜏_𝑛) for 𝑡∈[𝑡₀,𝑇]. Then, for 𝑡₀≤𝑡≤𝑇𝑥^𝑛(𝑡)=𝐺𝑥^𝑛_𝑡−𝐺(𝜉)+𝜉(0)+^𝑡_𝑡0𝑓𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}(𝑠)+d𝑠_𝑡𝑡0𝑔𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}(𝑠)d𝑊(𝑠)+^𝑡_𝑡0ℎ𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}(𝑠)d𝑁(𝑠), (3.9) where 𝐼_𝐴 is the indicator function of a set A, that is, 𝐼_𝐴(𝑥)=1,𝑥∈𝐴,0,𝑥∉𝐴. (3.10) Set 𝐽^𝑛(𝑡)=𝜉(0)+^𝑡_𝑡0𝑓𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}d𝑠+^𝑡_𝑡0𝑔𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}(𝑠)+d𝑊_𝑡𝑡0ℎ𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}d𝑁(𝑠). (3.11) Appling Lemma 3.4 twice and (3.4), one derives that 𝑥^𝑛(𝑡)²≤1𝜅𝐺𝑥^𝑛_𝑡−𝐺(𝜉)²+1𝐽1−𝜅^𝑛(𝑡)²≤1𝜅𝜅²𝑥^𝑛_𝑡−𝜉²+1𝐽1−𝜅^𝑛(𝑡)²=𝜅𝑥^𝑛_𝑡−𝜉²+1𝐽1−𝜅^𝑛(𝑡)²1≤𝜅√𝜅𝑥^𝑛_𝑡²+11−√𝜅𝜉²+1𝐽1−𝜅^𝑛(𝑡)²=√𝜅𝑥^𝑛_𝑡²+𝜅1−√𝜅𝜉²+1𝐽1−𝜅^𝑛(𝑡)². (3.12) Hence, 𝐸sup_{𝑡0≤𝑠≤𝑡}𝑥^𝑛(𝑠)²≤√𝜅𝐸sup_{𝑡0−𝜏≤𝑠≤𝑡}𝑥^𝑛(𝑠)²+𝜅1−√𝜅𝐸𝜉²+1𝐸1−𝜅sup_{𝑡0≤𝑠≤𝑡}𝐽^𝑛(𝑠)². (3.13) Noting that 𝐸(sup_{𝑡0−𝜏≤𝑠≤𝑡}|𝑥^𝑛(𝑠)|²)≤𝐸‖𝜉‖²+𝐸(sup_{𝑡0≤𝑠≤𝑡}|𝑥^𝑛(𝑠)|²), one sees that 𝐸sup_{𝑡0−𝜏≤𝑠≤𝑡}𝑥^𝑛(𝑠)²≤√𝜅𝐸sup_{𝑡0−𝜏≤𝑠≤𝑡}𝑥^𝑛(𝑠)²+1+𝜅−√𝜅1−√𝜅𝐸𝜉²+1𝐸1−𝜅sup_{𝑡0≤𝑠≤𝑡}𝐽^𝑛(𝑠)². (3.14) Consequently, 𝐸sup_{𝑡0−𝜏≤𝑠≤𝑡}𝑥^𝑛(𝑠)²≤1+𝜅−√𝜅1−√𝜅²𝐸𝜉²+1(1−𝜅)1−√𝜅𝐸sup_{𝑡0≤𝑠≤𝑡}𝐽^𝑛(𝑠)². (3.15) 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 𝐽^𝑛(𝑡)²≤4𝜉(0)²+4^𝑡_𝑡0𝑓𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}(𝑠)d𝑠²+4^𝑡_𝑡0𝑔𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}(𝑠)d𝑊(𝑠)²+4^𝑡_𝑡0ℎ𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}(𝑠)d𝑁(𝑠)²≤4𝜉(0)²+4𝑡−𝑡₀^𝑡_𝑡0𝑓𝑥^𝑛_𝑠,𝑠²d𝑠+4^𝑡_𝑡0𝑔𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}(𝑠)d𝑊(𝑠)²+4^𝑡_𝑡0ℎ𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}(𝑠)d𝑁(𝑠)+𝜆^𝑡_𝑡0ℎ𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}(𝑠)d𝑠²≤4𝜉(0)²+4𝑡−𝑡₀^𝑡_𝑡0𝑓𝑥^𝑛_𝑠,𝑠²d𝑠+4^𝑡_𝑡0𝑔𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}(𝑠)d𝑊(𝑠)²+8^𝑡_𝑡0ℎ𝑥^𝑛_𝑠,𝑠𝐼_{[𝑡0,𝜏𝑛]}(𝑠)d𝑁(𝑠)²+8𝜆²𝑡−𝑡₀^𝑡_𝑡0ℎ𝑥^𝑛_𝑠,𝑠²d𝑠. (3.16) Since both the Wiener process 𝑊(𝑡) and compensated Poisson process 𝑁(𝑡) are martingales, using the Doob martingale inequality, we have 𝐸sup_{𝑡0≤𝑠≤𝑡}𝐽^𝑛(𝑠)²≤4𝐸𝜉(0)²+4𝑡−𝑡₀𝐸^𝑡_𝑡0𝑓𝑥^𝑛_𝑠,𝑠²d𝑠+4×4𝐸^𝑡_𝑡0𝑔𝑥^𝑛_𝑠,𝑠²d𝑠+42×4𝜆+2𝜆²𝑡−𝑡₀𝐸^𝑡_𝑡0ℎ𝑥^𝑛_𝑠,𝑠²𝜉d𝑠≤4𝐸²+4𝐾𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀^𝑡_𝑡01+𝐸𝑥^𝑛_𝑠²d𝑠. (3.17) Substituting this into (3.15), 𝐸sup_{𝑡0−𝜏≤𝑠≤𝑡}𝑥^𝑛(𝑠)²≤5+𝜅√𝜅(1−𝜅)1−√𝜅𝐸𝜉²+4𝐾𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀(1−𝜅)1−√𝜅^𝑡_𝑡01+𝐸𝑥^𝑛_𝑠²d𝑠. (3.18) Therefore, 1+𝐸sup_{𝑡0−𝜏≤𝑠≤𝑡}𝑥^𝑛(𝑠)²√≤1+5+𝜅𝜅(1−𝜅)1−√𝜅𝐸𝜉²+4𝐾𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀(1−𝜅)1−√𝜅^𝑡_𝑡01+𝐸sup_{𝑡0−𝜏≤𝑟≤𝑠}𝑥^𝑛(𝑟)²d𝑠. (3.19) Now the Gronwall inequality yields that 1+𝐸sup_{𝑡0−𝜏≤𝑡≤𝑇}𝑥^𝑛(𝑡)²≤√1+5+𝜅𝜅(1−𝜅)1−√𝜅𝐸𝜉²×exp4𝐾𝑇−𝑡₀𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀(1−𝜅)1−√𝜅. (3.20) Consequently 𝐸sup_{𝑡0−𝜏≤𝑡≤𝜏𝑛}|𝑥(𝑡)|²≤√1+5+𝜅𝜅(1−𝜅)1−√𝜅𝐸𝜉²×exp4𝐾𝑇−𝑡₀𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀(1−𝜅)1−√𝜅. (3.21) Letting 𝑛→∞, it then implies the following inequality: 𝐸sup_{𝑡0−𝜏≤𝑡≤𝑇}|𝑥(𝑡)|²≤√1+5+𝜅𝜅(1−𝜅)1−√𝜅𝐸𝜉²×exp4𝐾𝑇−𝑡₀𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀(1−𝜅)1−√𝜅. (3.22) The proof is complete.

Proof of Theorem 3.2. 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];𝑅^𝑛). Note that 𝑥(𝑡)−𝑥(𝑡)=𝐺𝑥_𝑡−𝐺𝑥_𝑡+𝐽(𝑡), (3.23) where 𝐽(𝑡)=^𝑡_𝑡0𝑓𝑥_𝑠,𝑠−𝑓𝑥_𝑠,𝑠d𝑠+^𝑡_𝑡0𝑔𝑥_𝑠,𝑠−𝑔𝑥_𝑠(𝑠)+,𝑠d𝑊_𝑡𝑡0ℎ𝑥_𝑠,𝑠−ℎ𝑥_𝑠,𝑠d𝑁(𝑠). (3.24) By Lemma 3.4 and condition (3.4), one sees easily that |𝑥(𝑡)−𝑥(𝑡)|²≤1𝜅𝐺𝑥_𝑡−𝐺𝑥_𝑡²+1|𝐽(𝑡1−𝜅)|²≤𝜅𝑥_𝑡−𝑥_𝑡²+1|𝐽(𝑡1−𝜅)|². (3.25) Therefore, 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, (3.26) which implies that 𝐸sup_{𝑡0≤𝑠≤𝑡}|𝑥(𝑠)−𝑥(𝑠)|²≤1(1−𝜅)²𝐸sup_{𝑡0≤𝑠≤𝑡}|𝐽(𝑠)|². (3.27) On the other hand, 𝐸sup_{𝑡0≤𝑠≤𝑡}|𝐽(𝑠)|²≤3𝐾𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀^𝑡_𝑡0𝐸𝑥_𝑠−𝑥_𝑠²d𝑠=3𝐾𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀^𝑡_𝑡0𝐸sup_{𝑡0−𝜏≤𝑟≤𝑠}|𝑥(𝑟)−𝑥(𝑟)|²d𝑠. (3.28) Noting the fact that 𝑥(𝑟)=𝑥(𝑟)=𝜉, for 𝑟∈[𝑡₀−𝜏,𝑡₀], we have sup_{𝑡0−𝜏≤𝑟≤𝑠}|𝑥(𝑟)−𝑥(𝑟)|²=sup_{𝑡0≤𝑟≤𝑠}|𝑥(𝑟)−𝑥(𝑟)|². (3.29) Then 𝐸sup_{𝑡0≤𝑠≤𝑡}|𝐽(𝑠)|²≤3𝐾𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀^𝑡_𝑡0𝐸sup_{𝑡0≤𝑟≤𝑠}|𝑥(𝑟)−𝑥(𝑟)|²d𝑠. (3.30) Therefore, 𝐸sup_{𝑡0≤𝑠≤𝑡}|𝑥(𝑠)−𝑥(𝑠)|²≤3𝐾𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀(1−𝜅)^2𝑡_𝑡0𝐸sup_{𝑡0≤𝑟≤𝑠}|𝑥(𝑟)−𝑥(𝑟)|²d𝑠. (3.31) The Grownwall inequality then yields that 𝐸sup_{𝑡0≤𝑡≤𝑇}|𝑥(𝑡)−𝑥(𝑡)|²=0. (3.32) 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 3𝛿∶=𝜅+𝐾𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀1−𝜅𝑇−𝑡₀<1. (3.33) Define 𝑥⁰_𝑡0=𝜉 and 𝑥⁰(𝑡)=𝜉(0) for 𝑡₀≤𝑡≤𝑇. For each 𝑛=1,2,…, set 𝑥^𝑛_𝑡0=𝜉 and define, by the Picard iterations 𝑥^𝑛(𝑡)−𝐺𝑥_𝑡^𝑛−1=𝜉(0)−𝐺(𝜉)+^𝑡_𝑡0𝑓𝑥_𝑠^𝑛−1,𝑠d𝑠+^𝑡_𝑡0𝑔𝑥_𝑠^𝑛−1(𝑠)+,𝑠d𝑊_𝑡𝑡0ℎ𝑥_𝑠^𝑛−1,𝑠d𝑁(𝑠). (3.34) Obviously, 𝑥⁰(𝑡)∈ℳ²([𝑡₀−𝜏,𝑇];𝑅^𝑛), and by induction 𝑥^𝑛(𝑡)∈ℳ²([𝑡₀−𝜏,𝑇];𝑅^𝑛). In fact, 𝑥^𝑛(𝑡)=𝐺𝑥_𝑡^𝑛−1−𝐺(𝜉)+𝜉(0)+^𝑡_𝑡0𝑓𝑥_𝑠^𝑛−1,𝑠d𝑠+^𝑡_𝑡0𝑔𝑥_𝑠^𝑛−1(𝑠)+,𝑠d𝑊_𝑡𝑡0ℎ𝑥_𝑠^𝑛−1,𝑠d𝑁(𝑠). (3.35) Let 𝑃^𝑛−1(𝑡)=𝜉(0)+^𝑡_𝑡0𝑓𝑥_𝑠^𝑛−1,𝑠d𝑠+^𝑡_𝑡0𝑔𝑥_𝑠^𝑛−1,𝑠d𝑊(𝑠)+^𝑡_𝑡0ℎ𝑥_𝑠^𝑛−1,𝑠d𝑁(𝑠). (3.36) Then by Lemma 3.4 and (3.4) we have 𝑥^𝑛(𝑡)²≤1𝜅𝐺𝑥^𝑛_𝑡−𝐺(𝜉)²+1𝑃1−𝜅^𝑛−1(𝑡)²≤𝜅𝑥^𝑛_𝑡−𝜉²+1𝑃1−𝜅^𝑛−1(𝑡)²1≤𝜅√𝜅𝑥^𝑛_𝑡²+11−√𝜅𝜉²+1𝑃1−𝜅^𝑛−1(𝑡)²=√𝜅𝑥^𝑛_𝑡²+𝜅1−√𝜅𝜉²+1𝑃1−𝜅^𝑛−1(𝑡)²≤√𝜅𝜉²+𝑥^𝑛(𝑡)²+𝜅1−√𝜅𝜉²+1𝑃1−𝜅^𝑛−1(𝑡)². (3.37) Then, 𝑥^𝑛(𝑡)²≤√𝜅1−√𝜅²𝜉²+11−√𝜅()𝑃1−𝜅^𝑛−1(𝑡)². (3.38) Therefore, 𝐸𝑥^𝑛(𝑡)²≤√𝜅1−√𝜅²𝐸𝜉²+11−√𝜅()𝐸𝑃1−𝜅^𝑛−1(𝑡)². (3.39) 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 𝐸𝑃^𝑛−1(𝑡)²≤4𝐸𝜉(0)²+4𝐸^𝑡_𝑡0𝑓𝑥_𝑠^𝑛−1,𝑠d𝑠²+4𝐸^𝑡_𝑡0𝑔𝑥_𝑠^𝑛−1,𝑠d𝑊(𝑠)²+4𝐸^𝑡_𝑡0ℎ𝑥_𝑠^𝑛−1,𝑠d𝑁(𝑠)²≤4𝐸𝜉(0)²+4𝑡−𝑡₀𝐸^𝑡_𝑡0𝑓𝑥_𝑠^𝑛−1,𝑠²d𝑠+4𝐸^𝑡_𝑡0𝑔𝑥_𝑠^𝑛−1,𝑠²d𝑠+42𝜆+2𝜆²𝑡−𝑡₀𝐸^𝑡_𝑡0ℎ𝑥_𝑠^𝑛−1,𝑠²𝜉d𝑠≤4𝐸²+4𝐾𝑇−𝑡₀+1^𝑡_𝑡01+𝐸𝑥_𝑠^𝑛−12d𝑠+4𝐾2𝜆+2𝜆²𝑇−𝑡₀^𝑡_𝑡01+𝐸𝑥_𝑠^𝑛−12𝜉d𝑠≤4𝐸²+4𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀𝑇−𝑡₀+4𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀𝐸^𝑡_𝑡0sup_{𝑡0≤𝑟≤𝑠}𝑥^𝑛−1(𝑟)²𝜉d𝑠≤4𝐸²+4𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀𝑇−𝑡₀+4𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀^𝑡_𝑡0𝐸𝑥^𝑛−1(𝑠)²d𝑠. (3.40) Therefore, 𝐸𝑥^𝑛(𝑡)²≤√𝜅1−√𝜅²𝐸𝜉²+11−√𝜅()𝐸𝑃1−𝜅^𝑛−1(𝑡)²≤√𝜅1−√𝜅²𝐸𝜉²+41−√𝜅()𝐸𝜉1−𝜅²+4𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀𝑇−𝑡₀1−√𝜅()+1−𝜅4𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀1−√𝜅(1−𝜅)^𝑡_𝑡0𝐸𝑥^𝑛−1(𝑠)²d𝑠. (3.41) Hence, for any 𝑘≥1, we have max_{1≤𝑛≤𝑘}𝐸𝑥^𝑛(𝑡)²≤√𝜅1−√𝜅²+41−√𝜅()𝐸𝜉1−𝜅²+4𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀𝑇−𝑡₀1−√𝜅()+1−𝜅4𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀1−√𝜅(1−𝜅)^𝑡_𝑡0max_{1≤𝑛≤𝑘}𝐸𝑥^𝑛−1(𝑠)²d𝑠. (3.42) Note that max_{1≤𝑛≤𝑘}𝐸𝑥^𝑛−1(𝑡)²=max𝐸𝜉(0)²,𝐸𝑥¹(𝑠)²,…,𝐸𝑥^𝑘−1(𝑠)²≤max𝐸𝜉²,𝐸𝑥¹(𝑠)²,…,𝐸𝑥^𝑘−1(𝑠)²=max𝐸𝜉²,max_{1≤𝑛≤𝑘}𝐸𝑥^𝑛(𝑡)²≤𝐸𝜉²+max_{1≤𝑛≤𝑘}𝐸𝑥^𝑛(𝑡)². (3.43) Therefore, max_{1≤𝑛≤𝑘}𝐸𝑥^𝑛(𝑡)²≤√𝜅1−√𝜅²+41−√𝜅()𝐸𝜉1−𝜅²+4𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀𝑇−𝑡₀1−√𝜅()+1−𝜅4𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀1−√𝜅(1−𝜅)^𝑡_𝑡0𝐸𝜉²+max_{1≤𝑛≤𝑘}𝐸𝑥^𝑛(𝑡)²d𝑠≤𝑐₁+𝑐₂^𝑡_𝑡0max_{1≤𝑛≤𝑘}𝐸𝑥^𝑛(𝑠)²d𝑠, (3.44) 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 max_{1≤𝑛≤𝑘}𝐸𝑥^𝑛(𝑡)²≤𝑐₁𝑒^{𝑐2(𝑇−𝑡0)}. (3.45) Because 𝑘 is arbitrary, we have 𝐸𝑥^𝑛(𝑡)²≤𝑐₁𝑒^{𝑐2(𝑇−𝑡0)}. (3.46) Hence, for all 𝑛≥0, 𝑥^𝑛(𝑡)∈ℳ²([𝑡₀−𝜏,𝑇];𝑅^𝑛).
Note that, for 𝑡₀≤𝑡≤𝑇, 𝑥¹(𝑡)−𝑥⁰(𝑡)=𝑥¹(𝑡)(0)𝑥−𝜉=𝐺⁰_𝑡−𝐺(𝜉)+^𝑡_𝑡0𝑓𝑥⁰_𝑠,𝑠d𝑠+^𝑡_𝑡0𝑔𝑥⁰_𝑠(𝑠)+,𝑠d𝑊_𝑡𝑡0ℎ𝑥⁰_𝑠,𝑠d𝑁(𝑠). (3.47) In a similar way to that in the proof of the existence, we get that 𝐸sup_{𝑡0≤𝑡≤𝑇}𝑥¹(𝑡)−𝑥⁰(𝑡)²≤𝜅𝐸sup_{𝑡0≤𝑡≤𝑇}𝑥⁰_𝑡−𝜉²+3𝐾𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀1−𝜅^𝑡_𝑡01+𝑥⁰_𝑡²𝜉d𝑠≤2𝜅𝐸²+3𝐾𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀𝜉1−𝜅1+𝐸²𝑇−𝑡₀∶=𝐶. (3.48) Note also that, for 𝑛≥1 and 𝑡₀≤𝑡≤𝑇, 𝑥^𝑛+1(𝑡)−𝑥^𝑛(𝑡)=𝐺𝑥^𝑛_𝑡−𝐺𝑥_𝑡^𝑛−1+^𝑡_𝑡0𝑓𝑥^𝑛_𝑠,𝑠−𝑓𝑥_𝑠^𝑛−1+,𝑠d𝑠_𝑡𝑡0𝑔𝑥^𝑛_𝑠,𝑠−𝑔𝑥_𝑠^𝑛−1(𝑠)+,𝑠d𝑊_𝑡𝑡0ℎ𝑥^𝑛_𝑠,𝑠−ℎ𝑥_𝑠^𝑛−1,𝑠d𝑁(𝑠). (3.49) In the same way as in the proof of the uniqueness, one derives that 𝐸sup_{𝑡0≤𝑡≤𝑇}𝑥^𝑛+1(𝑡)−𝑥^𝑛(𝑡)²≤𝜅𝐸sup_{𝑡0≤𝑡≤𝑇}𝑥^𝑛(𝑡)−𝑥^𝑛−1(𝑡)²+3𝐾𝑇−𝑡₀+4+8𝜆+2𝜆²𝑇−𝑡₀1−𝜅^𝑇_𝑡0𝐸sup_{𝑡0≤𝑠≤𝑡}𝑥^𝑛(𝑠)−𝑥^𝑛−1(𝑠)²d𝑡≤𝛿𝐸sup_{𝑡0≤𝑡≤𝑇}𝑥^𝑛(𝑡)−𝑥^𝑛−1(𝑡)²≤𝛿^𝑛𝐸sup_{𝑡0≤𝑡≤𝑇}𝑥¹(𝑡)−𝑥⁰(𝑡)²≤𝐶𝛿^𝑛. (3.50) Next we verify that {𝑥^𝑛(𝑡)} converges to 𝑥(𝑡) in the sense of 𝐿² and probability 1 on ℳ²([𝑡₀−𝜏,𝑇];𝑅^𝑛). Moreover, 𝑥(𝑡) is the solution to (2.1) with initial data (2.3). Hence, by the Chebyshev inequality, 𝑃sup_{𝑡0≤𝑡≤𝑇}𝑥^𝑛+1(𝑡)−𝑥^𝑛(𝑡)²>12^𝑛≤𝐶(4𝛿)^𝑛. (3.51) Since ∑^∞_𝑛=0𝐶(4𝛿)^𝑛≤∞, the Borel-Cantelli lemma yields that, for almost all 𝜔∈Ω, there exist a positive integer 𝑛₀=𝑛₀(𝜔) such that sup_{𝑡0≤𝑡≤𝑇}𝑥^𝑛+1(𝑡)−𝑥^𝑛(𝑡)²≤12^𝑛,𝑛≥𝑛₀. (3.52) It follows that, with probability 1, the partial sum 𝑥⁰(𝑡)+^𝑛−1_𝑖=0𝑥^𝑖+1(𝑡)−𝑥^𝑖(𝑡)=𝑥^𝑛(𝑡) (3.53) is the partial sum of function series 𝑥⁰(𝑡)+𝑥¹(𝑡)−𝑥⁰(𝑡)+⋯+𝑥^𝑛(𝑡)−𝑥^𝑛−1(𝑡)+⋯. (3.54) By the second item of series (3.54), the absolute value of every item (3.54) is less than the corresponding item of positive 11+2+12²1+⋯+2^𝑛+⋯. (3.55) 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 {𝑥^𝑛(𝑡)} uniformly converges to 𝑥(𝑡) on [𝑡₀,𝑇] and is ℱ_𝑡-adapted, hence 𝑥(𝑡) is also continuous and ℱ_𝑡-adapted. On the other hand, (3.50) implies that, for each 𝑡, sequence {𝑥^𝑛(𝑡)} is a Cauchy sequence in ℳ²([𝑡₀−𝜏,𝑇];𝑅^𝑛) as well. Hence, we also have 𝑥^𝑛(𝑡)→𝑥(𝑡) in ℳ²([𝑡₀−𝜏,𝑇];𝑅^𝑛). Letting 𝑛→∞ in (3.46) gives 𝐸|𝑥(𝑡)|²≤𝑐₁𝑒^{𝑐2(𝑇−𝑡0)},𝑡₀≤𝑡≤𝑇. (3.56) Therefore, by the use of the above result, we get that 𝐸^𝑇_𝑡0−𝜏|𝑥(𝑠)|²d𝑠=𝐸^𝑡0_𝑡0−𝜏|𝑥(𝑠)|²d𝑠+𝐸^𝑇_𝑡0|𝑥(𝑠)|²d𝑠≤𝐸^𝑡0_𝑡0−𝜏𝜉(𝑠)²d𝑠+𝐸^𝑇_𝑡0−𝜏𝑐₁𝑒^{𝑐2(𝑇−𝑡0)}d𝑠<∞. (3.57) That is, 𝑥(𝑡)∈ℳ²([𝑡₀−𝜏,𝑇];𝑅^𝑛).
Now we prove that 𝑥(𝑡) satisfies (2.1): 𝐸^𝑡_𝑡0𝑓𝑥^𝑛_𝑠,𝑠−𝑓𝑥_𝑠,𝑠d𝑠²+𝐸^𝑡_𝑡0𝑔𝑥^𝑛_𝑠,𝑠−𝑔𝑥_𝑠,𝑠d𝑊(𝑠)²+𝐸^𝑡_𝑡0ℎ𝑥^𝑛_𝑠,𝑠−ℎ𝑥_𝑠,𝑠d𝑁(𝑠)²≤𝑡−𝑡₀𝐸^𝑡_𝑡0𝑓𝑥^𝑛_𝑠,𝑠−𝑓𝑥_𝑠,𝑠²d𝑠+𝐸^𝑡_𝑡0𝑔𝑥^𝑛_𝑠,𝑠−𝑔𝑥_𝑠,𝑠²+d𝑠2𝜆+2𝜆²𝑡−𝑡₀𝐸^𝑡_𝑡0ℎ𝑥^𝑛_𝑠,𝑠−ℎ𝑥_𝑠,𝑠²≤d𝑠𝑡−𝑡₀𝐾𝐸^𝑡_𝑡0𝑥^𝑛_𝑠−𝑥_𝑠²d𝑠+𝐾𝐸^𝑡_𝑡0𝑥^𝑛_𝑠−𝑥_𝑠²+d𝑠2𝜆+2𝜆²𝑡−𝑡₀𝐾𝐸^𝑡_𝑡0𝑥^𝑛_𝑠−𝑥_𝑠≤d𝑠𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀𝐸^𝑡_𝑡0𝑥^𝑛_𝑠−𝑥_𝑠≤d𝑠𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀^𝑡_𝑡0𝐸sup_{𝑡0≤𝑟≤𝑠}𝑥^𝑛(𝑟)−𝑥(𝑟)²≤d𝑠𝐾𝑇−𝑡₀+1+2𝜆+2𝜆²𝑇−𝑡₀^𝑇_𝑡0𝐸𝑥^𝑛(𝑠)−𝑥(𝑠)²().d𝑠⟶0𝑛⟶∞ (3.58) Hence, we can let 𝑛→∞ in (3.34) to obtain that 𝑥(𝑡)−𝐺𝑥_𝑡=𝜉(0)−𝐺(𝜉)+^𝑡_𝑡0𝑓𝑥_𝑠,𝑠d𝑠+^𝑡_𝑡0𝑔𝑥_𝑠(𝑠)+,𝑠d𝑊_𝑡𝑡0ℎ𝑥_𝑠,𝑠d𝑁(𝑠). (3.59) 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 3𝛿∶=𝜅+𝐾𝜎+4+8𝜆+2𝜆²𝜎1−𝜅𝜎<1. (3.60) 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 𝑥_𝑡0+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 Theorem 3.2 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 constants 𝐾_𝑛 and 𝐾 such that(i)(the local Lipschitz condition) for all 𝜙,𝜑∈𝐶([−𝜏,0];𝑅^𝑛) with ‖𝜙‖∨|𝜑‖≤𝑛 and 𝑡∈[𝑡₀,𝑇], 𝑓𝜙,𝑡−𝑓𝜑,𝑡²∨𝑔𝜙,𝑡−𝑔𝜑,𝑡²∨ℎ𝜙,𝑡−ℎ𝜑,𝑡²≤𝐾_𝑛𝜙−𝜑². (3.61)(ii)(the linear growth condition) for all (𝜙,𝑡)∈𝐶([−𝜏,0];𝑅^𝑛)×[𝑡₀,𝑇], 𝑓𝜙,𝑡²∨𝑔𝜙,𝑡²∨ℎ𝜙,𝑡²≤𝐾1+𝜙². (3.62)(iii)(the constractive mapping) there is a positive constant 𝜅∈(0,1) such that, for all 𝜙,𝜑∈𝐶([−𝜏,0];𝑅^𝑛), 𝐺𝜙−𝐺𝜑≤𝜅𝜙−𝜑. (3.63)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];𝑅^𝑛).

Proof. For each 𝑛≥1, define truncation functions 𝑓_𝑛,𝑔_𝑛, and ℎ_𝑛 as follows: 𝑓_𝑛𝑥_𝑡=𝑓𝑥,𝑡^𝑡,𝑡𝑥_𝑡𝑓≤𝑛,𝑛𝑥_𝑡𝑥_𝑡,𝑡𝑥_𝑡𝑔>𝑛,_𝑛𝑥_𝑡=𝑔𝑥,𝑡^𝑡,𝑡𝑥_𝑡𝑔≤𝑛,𝑛𝑥_𝑡𝑥_𝑛,𝑡𝑥_𝑡ℎ>𝑛,_𝑛𝑥_𝑡=ℎ𝑥,𝑡^𝑡,𝑡𝑥_𝑡ℎ≤𝑛,𝑛𝑥_𝑡𝑥_𝑛,𝑡𝑥_𝑡>𝑛, (3.64) Then 𝑓_𝑛, 𝑔_𝑛, and ℎ_𝑛 satisfy conditions (3.2) and (3.3). By Theorem 3.2, there is a unique solution 𝑥^𝑛(𝑡) and 𝑥^𝑛(𝑡)∈ℳ²([𝑡₀−𝜏,𝑇];𝑅^𝑛) to the equation 𝑥^𝑛(𝑡)−𝐺𝑥^𝑛_𝑡=𝜉(0)−𝐺(𝜉)+^𝑡_𝑡0𝑓_𝑛𝑥^𝑛_𝑠+,𝑠d𝑠_𝑡𝑡0𝑔_𝑛𝑥^𝑛_𝑠(𝑠)+,𝑠d𝑊_𝑡𝑡0ℎ_𝑛𝑥^𝑛_𝑠,𝑠d𝑁(𝑠),𝑡∈𝑡₀,𝑇. (3.65) Certainly 𝑥^𝑛+1(𝑡) is a unique solution to the equation 𝑥^𝑛+1(𝑡)−𝐺𝑥_𝑡^𝑛+1=𝜉(0)−𝐺(𝜉)+^𝑡_𝑡0𝑓_𝑛+1𝑥_𝑠^𝑛+1+,𝑠d𝑠_𝑡𝑡0𝑔_𝑛+1𝑥_𝑠^𝑛+1(𝑠)+,𝑠d𝑊_𝑡𝑡0ℎ_𝑛+1𝑥_𝑠^𝑛+1,𝑠d𝑁(𝑠),𝑡∈𝑡₀,𝑇 (3.66) and 𝑥^𝑛+1(𝑡)∈ℳ²([𝑡₀−𝜏,𝑇];𝑅^𝑛) to the equation.
Now define the stopping time 𝜏_𝑛=𝑇∧inf𝑡∈𝑡₀,𝑇∶𝑥^𝑛_𝑡≥𝑛. (3.67) We can show that 𝑥^𝑛(𝑡)=𝑥^𝑛+1(𝑡) for 𝑡∈[𝑡₀,𝑇]. Then by Lemma 3.4, basic inequality, we have 𝑥^𝑛+1(𝑡)−𝑥^𝑛(𝑡)²≤1𝜅𝐺(𝑥_𝑡^𝑛+1)−𝐺(𝑥^𝑛_𝑡)²+31−𝜅^𝑡_𝑡0𝑓_𝑛+1𝑥_𝑠^𝑛+1,𝑠−𝑓_𝑛𝑥^𝑛_𝑠,𝑠d𝑠²+31−𝜅^𝑡_𝑡0𝑔_𝑛+1𝑥_𝑠^𝑛+1,𝑠−𝑔_𝑛𝑥^𝑛_𝑠,𝑠d𝑊(𝑠)²+31−𝜅^𝑡_𝑡0ℎ_𝑛+1𝑥_𝑠^𝑛+1,𝑠−ℎ_𝑛𝑥^𝑛_𝑠,𝑠d𝑁(𝑠)². (3.68)
Taking the expectation, by the Hölder inequality and Lemma 3.3, we transform the Poisson process to the compensated Poisson process 𝑁(𝑡)=𝑁(𝑡)−𝜆𝑡, and, by martingale isometry, we have 𝐸𝑥^𝑛+1(𝑡)−𝑥^𝑛(𝑡)²≤𝜅𝐸𝑥_𝑡^𝑛+1−𝑥^𝑛_𝑡²+31−𝜅𝑡−𝑡₀𝐸^𝑡_𝑡0𝑓_𝑛+1𝑥_𝑠^𝑛+1,𝑠−𝑓_𝑛𝑥^𝑛_𝑠,𝑠²+3d𝑠𝐸1−𝜅^𝑡_𝑡0𝑔_𝑛+1𝑥_𝑠^𝑛+1,𝑠−𝑔_𝑛𝑥^𝑛_𝑠,𝑠²+3d𝑠1−𝜅2𝜆+2𝜆²𝑡−𝑡₀𝐸^𝑡_𝑡0ℎ_𝑛+1𝑥_𝑠^𝑛+1,𝑠−ℎ_𝑛𝑥^𝑛_𝑠,𝑠²𝑥d𝑠≤𝜅𝐸_𝑡^𝑛+1−𝑥^𝑛_𝑡²+61−𝜅𝑡−𝑡₀𝐸^𝑡_𝑡0𝑓_𝑛+1𝑥_𝑠^𝑛+1,𝑠−𝑓_𝑛+1𝑥^𝑛_𝑠,𝑠²+6d𝑠1−𝜅𝑡−𝑡₀𝐸^𝑡_𝑡0𝑓_𝑛+1𝑥^𝑛_𝑠,𝑠−𝑓_𝑛𝑥^𝑛_𝑠,𝑠²+6d𝑠𝐸1−𝜅^𝑡_𝑡0𝑔_𝑛+1𝑥_𝑠^𝑛+1,𝑠−𝑔_𝑛+1𝑥^𝑛_𝑠,𝑠²+6d𝑠1−𝜅𝐸^𝑡_𝑡0𝑔_𝑛+1𝑥^𝑛_𝑠,𝑠−𝑔_𝑛𝑥^𝑛_𝑠,𝑠²+6d𝑠1−𝜅2𝜆+2𝜆²𝑡−𝑡₀𝐸^𝑡_𝑡0ℎ_𝑛+1𝑥_𝑠^𝑛+1,𝑠−ℎ_𝑛+1𝑥^𝑛_𝑠,𝑠²+6d𝑠1−𝜅2𝜆+2𝜆²𝑡−𝑡₀𝐸^𝑡_𝑡0ℎ_𝑛+1𝑥^𝑛_𝑠,𝑠−ℎ_𝑛𝑥^𝑛_𝑠,𝑠²d𝑠. (3.69) For 𝑡₀<𝑡<𝑡_𝑛, we have know that 𝑓_𝑛+1𝑥^𝑛_𝑠,𝑠=𝑓_𝑛𝑥^𝑛_𝑠,𝑠=𝑓𝑥^𝑛_𝑠,𝑔,𝑠_𝑛+1𝑥^𝑛_𝑠,𝑠=𝑔_𝑛𝑥^𝑛_𝑠,𝑠=𝑔𝑥^𝑛_𝑠,ℎ,𝑠_𝑛+1𝑥^𝑛_𝑠,𝑠=ℎ_𝑛𝑥^𝑛_𝑠,𝑠=ℎ𝑥^𝑛_𝑠,𝑠. (3.70) Again by 𝑥_𝑡^𝑛+1₀=𝑥^𝑛_𝑡0=𝜉, we get that 𝐸𝑥^𝑛+1(𝑠)−𝑥^𝑛(𝑠)²≤𝜅𝐸𝑥_𝑡^𝑛+1−𝑥^𝑛_𝑡²+61−𝜅𝑡−𝑡₀𝐸^𝑡_𝑡0𝑓_𝑛+1𝑥_𝑠^𝑛+1,𝑠−𝑓_𝑛+1𝑥^𝑛_𝑠,𝑠²+6d𝑠𝐸1−𝜅^𝑡_𝑡0𝑔_𝑛+1𝑥_𝑠^𝑛+1,𝑠−𝑔_𝑛+1𝑥^𝑛_𝑠,𝑠²+6d𝑠1−𝜅2𝜆+2𝜆²𝑡−𝑡₀𝐸^𝑡_𝑡0ℎ_𝑛+1𝑥_𝑠^𝑛+1,𝑠−ℎ_𝑛+1𝑥^𝑛_𝑠,𝑠²d𝑠,=𝜅𝐸sup_{𝑡0−𝜏≤𝑠≤𝑡}𝑥^𝑛+1(𝑠)−𝑥^𝑛(𝑠)²+61−𝜅𝑡−𝑡₀𝐸^𝑡_𝑡0𝑓𝑥_𝑠^𝑛+1,𝑠−𝑓𝑥^𝑛_𝑠,𝑠²+6d𝑠𝐸1−𝜅^𝑡_𝑡0𝑔𝑥_𝑠^𝑛+1,𝑠−𝑔𝑥^𝑛_𝑠,𝑠²+6d𝑠1−𝜅2𝜆+2𝜆²𝑡−𝑡₀𝐸^𝑡_𝑡0ℎ𝑥_𝑠^𝑛+1,𝑠−ℎ𝑥^𝑛_𝑠,𝑠²d𝑠. (3.71) Noting the fact that 𝑥^𝑛+1(𝑠)=𝑥^𝑛(𝑠)=𝜉, for 𝑠∈[𝑡₀−𝜏,𝑡₀], we get 𝐸sup_{𝑡0≤𝑠≤𝑡}𝑥^𝑛+1(𝑠)−𝑥^𝑛(𝑠)²≤6𝑡−𝑡₀+1+2𝜆+2𝜆²𝑡−𝑡₀(1−𝜅)²𝐸^𝑡_𝑡0𝐾_𝑛𝑥_𝑠^𝑛+1−𝑥^𝑛_𝑠²≤6d𝑠𝑡−𝑡₀+1+2𝜆+2𝜆²𝑡−𝑡₀(1−𝜅)²𝐾_𝑛𝐸^𝑡_𝑡0sup_{𝑡0≤𝑠≤𝑡}𝑥^𝑛+1(𝑠)−𝑥^𝑛(𝑠)²d𝑠. (3.72) From the Gronwall inequality one sees that 𝐸sup_{𝑡0≤𝑠≤𝑡}𝑥^𝑛+1(𝑠)−𝑥^𝑛(𝑠)²,𝑡₀<𝑡<𝜏_𝑛. (3.73) This means that, for 𝑡₀<𝑡<𝜏_𝑛, we always have 𝑥^𝑛(𝑡)=𝑥^𝑛+1(𝑡). (3.74) It is then deduced that 𝜏_𝑛 is increasing, that is as 𝑛→∞, 𝜏_𝑛↑𝑇 a.s. By the linear growth condition, for all almost all 𝜔∈Ω, there exists an integer 𝑛₀=𝑛₀(𝜔) such that 𝜏_𝑛=𝑇 as 𝑛>𝑛₀. Now define 𝑥(𝑡) by 𝑥(𝑡)=𝑥_𝑛0(𝑡), 𝑡∈[𝑡₀,𝑇]. Next to verify that 𝑥(𝑡) is the solution to (2.1), by (3.74) 𝑥(𝑡∧𝜏_𝑛)=𝑥_𝑛(𝑡∧𝜏_𝑛), it follows that 𝑥𝑡∧𝜏_𝑛−𝐺𝑥^𝑛_{𝑡∧𝜏𝑛}=𝜉(0)−𝐺(𝜉)+^{𝑡∧𝜏𝑛}_𝑡0𝑓_𝑛𝑥_𝑠+,𝑠d𝑠_{𝑡∧𝜏𝑛𝑡0}𝑔_𝑛𝑥_𝑠,𝑠d𝑊(𝑠)+^{𝑡∧𝜏𝑛}_𝑡0ℎ_𝑛𝑥_𝑠,𝑠d𝑁(𝑠), (3.75) that is, 𝑥𝑡∧𝜏_𝑛−𝐺𝑥^𝑛_{𝑡∧𝜏𝑛}=𝜉(0)−𝐺(𝜉)+^{𝑡∧𝜏𝑛}_𝑡0𝑓𝑥_𝑠+,𝑠d𝑠_{𝑡∧𝜏𝑛𝑡0}𝑔𝑥_𝑠,𝑠d𝑊(𝑠)+^{𝑡∧𝜏𝑛}_𝑡0ℎ𝑥_𝑠,𝑠d𝑁(𝑠). (3.76) Letting 𝑛→∞𝑥(𝑡∧𝑇)−𝐺𝑥^𝑛_𝑡∧𝑇=𝜉(0)−𝐺(𝜉)+_𝑡^𝑡∧𝑇₀𝑓𝑥_𝑠+,𝑠d𝑠_{𝑡𝑡∧𝑇0}𝑔𝑥_𝑠,𝑠d𝑊(𝑠)+_𝑡^𝑡∧𝑇₀ℎ𝑥_𝑠,𝑠d𝑁(𝑠), (3.77) that is 𝑥(𝑡)−𝐺𝑥_𝑡=𝜉(0)−𝐺(𝜉)+^𝑡_𝑡0𝑓𝑥_𝑠+,𝑠d𝑠_𝑡𝑡0𝑔𝑥_𝑠,𝑠d𝑊(𝑠)+^𝑡_𝑡0ℎ𝑥_𝑠,𝑠d𝑁(𝑠). (3.78) We can see that 𝑥(𝑡) is the solution to (2.1) and 𝑥(𝑡)∈ℳ²([𝑡₀−𝜏,𝑇];𝑅^𝑛). The proof of existence is complete. By stopping our process, uniqueness is obtained. This completes the proof.

Acknowledgments

This research was supported with funds provided by the National Natural Science Foundation of China (no. 11102132 and no. 10732020). The authors thank two anonymous reviewers for their very valuable comments and helpful suggestions which improve this paper significantly.