Reflected Backward Stochastic Differential Equations Driven by Countable Brownian Motions

Abstract

This paper deals with a new class of reflected backward stochastic differential equations driven by countable Brownian motions. The existence and uniqueness of the RBSDEs are obtained via Snell envelope and fixed point theorem.

1. Introduction

The nonlinear backward stochastic differential equations (BSDEs in short) were introduced by Pardoux and Peng [1], who proved the existence and uniqueness of the solution under the Lipschitz conditions for giving the probabilistic interpretation of semilinear parabolic partial differential equations. Since then, many authors were devoted to studying the BSDEs (see, e.g., [2–8] and the references therein). At present, the theory of BSDEs becomes a powerful tool to solve practical matters. In 1994, Pardoux and Peng [9] firstly studied the backward doubly stochastic differential equations (BDSDEs in short), which are driven by two kinds of Brownian motions. Later, Boufoussi et al. [10] established the connection between a class of generalized BDSDEs and semilinear stochastic partial differential equations with a Neumann boundary condition.

Reflected backward differential equations (RBSDEs in short) were introduced by El Karoui et al. [11]. Later, many researchers discussed various kinds of RBSDEs for their deep application in mathematical finance and partial differential equations. Ren and Hu [12] proposed the RBSDEs, driven by Teugels martingales and Brownian motion, and derived the existence and uniqueness of the solution by means of the Snell envelope and the fixed point theorem when the barrier was right continuous with left limits. Ren and El Otmani [13] discussed the generalized reflected BSDEs driven by Lévy process. Recently, Ren et al. [14] studied a new class of reflected backward doubly stochastic differential equations driven by Lévy process and Brownian motion.

As in all the previous works, the equations are driven by finite Brownian motions. To the best of our knowledge, there are no papers on the reflected backward stochastic differential equations driven by countable Brownian motions. In this paper, we aim to derive the existence and uniqueness of the solution for the RBSDEs driven by countable Brownian motions.

The structure of the paper is organized as follows. In Section 2, we give some notations. Section 3 is devoted to the main result.

2. Notations

Let 𝑇 be a positive constant. Throughout the paper (Ω,ℱ,ℙ) is a complete probability space equipped with the natural filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions. {𝛽_𝑗(𝑡)}^∞_𝑗=1 are mutual independent one-dimensional standard Brownian motions on the probability space. 𝑊(𝑡) is a standard Brownian motion on ℝ^𝑑 which is independent of 𝛽_𝑗(𝑡). Assume that ℱ_𝑡=^∞_𝑗=1ℱ^𝛽𝑗_𝑡,𝑇ℱ^𝑊_𝑡𝒩, (1) where for any process {𝜂_𝑡}, ℱ^𝜂_𝑠,𝑡=𝜎{𝜂_𝑟−𝜂_𝑠∶𝑠≤𝑟≤𝑡}, ℱ^𝜂_𝑡=ℱ^𝜂_0,𝑡, and 𝒩 denotes the class of ℙ-null sets of ℱ.

For the convenience, let us introduce some spaces: (i) ℋ²={(𝜑_𝑡)_{0≤𝑡≤𝑇}: an ℱ_𝑡-progressively measurable, ℝ-valued process such that 𝐸∫^𝑇₀|𝜑_𝑡|²d𝑡<∞};(ii) 𝒮²={(𝜓_𝑡)_{0≤𝑡≤𝑇}: an ℱ_𝑡-progressively measurable, ℝ^𝑑-valued continuous process such that 𝐸(sup_{0≤𝑡≤𝑇}|𝜓_𝑡|²)<∞};(iii) 𝒜²={(𝐾_𝑡)_{0≤𝑡≤𝑇}: an ℱ_𝑡-adapted, continuous, increasing process such that 𝐾₀=0,𝐸|𝐾_𝑡|²<∞}.

With the previous preparations, we consider the following RBSDEs: 𝑌_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝑌_𝑠,𝑍_𝑠+d𝑠_{∞𝑗=1𝑇𝑡}𝑔_𝑗𝑠,𝑌_𝑠,𝑍_𝑠d𝛽_𝑗(𝑠)−^𝑇_𝑡𝑍_𝑠d𝑊(𝑠)+𝐾_𝑇−𝐾_𝑡,0≤𝑡≤𝑇, (2) where 𝑓∶Ω×[0,𝑇]×ℝ×ℝ^𝑑→ℝ  and  𝑔_𝑗∶Ω×[0,𝑇]×ℝ×ℝ^𝑑→ℝ.

Definition 1. A solution of (2) is a triple of ℝ×ℝ^𝑑×ℝ₊ value process (𝑌_𝑡,𝑍_𝑡,𝐾_𝑡)_{0≤𝑡≤𝑇}, which satisfies (2), and (i)(𝑌_𝑡,𝑍_𝑡,𝐾_𝑡)_{0≤𝑡≤𝑇}∈𝒮²×ℋ²×𝒜²; (ii)𝑌_𝑡≥𝑆_𝑡; (iii)𝐾_𝑡 is a continuous and increasing process with 𝐾₀=0 and ∫^𝑇₀(𝑌_𝑡−𝑆_𝑡)d𝐾_𝑡=0.
In order to get the solution of (2), we propose the following assumptions: (H1)𝜉 is an ℱ_𝑇 measurable square integrable random variable; (H2) the obstacle {𝑆_𝑡∶0≤𝑡≤𝑇} is an ℱ_𝑡-progressive measurable continuous real valued process which satisfies 𝐸sup_{0≤𝑡≤𝑇}(𝑆_𝑡)²<∞. We always assume that 𝑆_𝑇≤𝜉, a.s.; (H3)𝑓(⋅,𝑦,𝑧) and 𝑔_𝑗(⋅,𝑦,𝑧) are two progressive measurable functions such that, for any 𝑡∈[0,𝑇], 𝑦₁,𝑦₂∈ℝ, 𝑧₁,𝑧₂∈ℝ^𝑑, (3a)𝑓(𝑠,⋅,⋅) is continuous and |𝑓(𝑠,𝑦,𝑧)|≤𝑀(1+|𝑦|+|𝑧|); (3b)𝐸∫^𝑇₀|𝑓(𝑡,0,0)|²dt<∞, ∑^∞_𝑗=1𝐸∫^𝑇₀|𝑔_𝑗(𝑡,0,0)|²dt<∞; (3c)|𝑓(𝑠,𝑦₁,𝑧₁)−𝑓(𝑠,𝑦₂,𝑧₂)|²≤𝐶(|𝑦₁−𝑦₂|²+|𝑧₁−𝑧₂|²), |𝑔_𝑗(𝑠,𝑦₁,𝑧₁)−𝑔_𝑗(𝑠,𝑦₂,𝑧₂)|²≤𝐶_𝑗|𝑦₁−𝑦₂|²+𝛼_𝑗|𝑧₁−𝑧₂|², where 𝑀, 𝐶, 𝐶_𝑗,   and  𝛼_𝑗 are nonnegative constants with ∑^∞_𝑗=1𝐶_𝑗<∞ and 𝛼=∑^∞_𝑗=1𝛼_𝑗<1.

3. Main Result

In order to get the solution of (2), we consider the following RBSDEs driven by finite Brownian motions: 𝑌_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝑌_𝑠,𝑍_𝑠+d𝑠_{𝑛𝑗=1𝑇𝑡}𝑔_𝑗𝑠,𝑌_𝑠,𝑍_𝑠d𝛽_𝑗(𝑠)−^𝑇_𝑡𝑍_𝑠d𝑊(𝑠)+𝐾_𝑇−𝐾_𝑡,0≤𝑡≤𝑇. (3)

Firstly, we consider a special case of (3); that is, the functions 𝑓 and 𝑔 do not depend on (𝑌,𝑍): 𝑌_𝑡=𝜉+^𝑇_𝑡𝑓(𝑠)+d𝑠_{𝑛𝑗=1𝑇𝑡}𝑔_𝑗(𝑠)d𝛽_𝑗(𝑠)−^𝑇_𝑡𝑍_𝑠d𝑊(𝑠)+𝐾_𝑇−𝐾_𝑡,0≤𝑡≤𝑇,𝑛≥1. (4)

We will get the existence and uniqueness of the solution of (4) by means of Snell envelope and martingale representation theorem.

Theorem 2. Assume that (H1)-(H2), 𝑓∈ℋ², 𝑔∈ℋ². Then, there exists a triple (𝑌_𝑡,𝑍_𝑡,𝐾_𝑡)_{0≤𝑡≤𝑇}∈𝒮²×ℋ²×𝒜² which is a solution of (4).

Proof. Let 𝒞_𝑡=ℱ^𝑊_𝑡^𝑛_𝑗=1ℱ^𝛽𝑗_𝑡,𝑇, (5) and we define 𝜂={𝜂_𝑡}_{0≤𝑡≤𝑇} as 𝜂_𝑡=𝜉1_{𝑡=𝑇}+𝑆_𝑡1_{𝑡<𝑇}+^𝑡₀𝑓(𝑠)+d𝑠_{𝑛𝑗=1𝑡0}𝑔_𝑗(𝑠)d𝛽_𝑗(𝑠). (6) Then, 𝜂 is 𝒞_𝑡-adapted continuous process; furthermore; sup_{0≤𝑡≤𝑇}𝜂_𝑡∈𝐿²(Ω). (7) So, the Snell envelope of 𝜂 is given by 𝑆_𝑡𝜂=esssup_𝜈∈𝒯𝐸𝜂_𝜈∣𝒞_𝑡, (8) where 𝒯 is the set of all 𝒞_𝑡 stopping time such that 0≤𝜈≤𝑇.
By the definition of 𝜂, we can deduce that 𝐸sup_{0≤𝑡≤𝑇}𝑆_𝑡𝜂²<∞. (9)
Due to the Doob-Meyer decomposition, we have 𝑆_𝑡𝜂=𝐸𝜉+^𝑇₀𝑓(𝑠)+d𝑠_{𝑛𝑗=1𝑇0}𝑔_𝑗(𝑠)d𝛽_𝑗(𝑠)+𝐾_𝑇∣𝒞_𝑡−𝐾_𝑡, (10) where {𝐾_𝑡}_{0≤𝑡≤𝑇} is a 𝒞_𝑡-adapted, continuous, and nondecreasing process such that 𝐾₀=0 and 𝐸𝐾²_𝑇<∞. So, we have 𝐸sup_{0≤𝑡≤𝑇}𝐸𝜉+^𝑇₀𝑓(𝑠)+d𝑠_{𝑛𝑗=1𝑇0}𝑔_𝑗(𝑠)d𝛽_𝑗(𝑠)+𝐾_𝑇∣𝒞_𝑡²<∞. (11)
Martingale representation theorem yields that there exists 𝒞_𝑡-progressive measurable process {𝑍_𝑡}∈ℝ^𝑑 such that 𝑀_𝑡≜𝐸𝜉+^𝑇₀𝑓(𝑠)d𝑠+^𝑛_𝑗=1^𝑇₀𝑔_𝑗(𝑠)d𝛽_𝑗(𝑠)+𝐾_𝑇∣𝒞_𝑡=𝑀₀+^𝑡₀𝑍_𝑠d𝑊(𝑠),0≤𝑡≤𝑇. (12)
Let 𝑌_𝑡=esssup_𝜈∈𝒯𝐸[𝜉1_{𝜈=𝑇}+𝑆_𝜈1_{𝜈<𝑇}+∫^𝜈_𝑡𝑓(𝑠)d𝑠+∑^𝑛_𝑗=1∫^𝜈_𝑡𝑔_𝑗(𝑠)d𝛽_𝑗(𝑠)∣𝒞_𝑡]; then, 𝑌_𝑡+^𝑡₀𝑓(𝑠)d𝑠+^𝑛_𝑗=1^𝑡₀𝑔_𝑗(𝑠)d𝛽_𝑗(𝑠)=𝑆_𝑡𝜂=𝑀_𝑡−𝐾_𝑡=𝑀₀+^𝑡₀𝑍_𝑠d𝑊(𝑠)−𝐾_𝑡,0≤𝑡≤𝑇. (13) Therefore, 𝑌_𝑡=𝜉+^𝑇_𝑡𝑓(𝑠)d𝑠+^𝑛_𝑗=1^𝑇_𝑡𝑔_𝑗(𝑠)d𝛽_𝑗(𝑠)−^𝑇_𝑡𝑍_𝑠d𝑊(𝑠)+𝐾_𝑇−𝐾_𝑡. (14)
By the definitions of 𝑌_𝑡 and 𝑆_𝑡(𝜂), 𝜉≥𝑆_𝑇, 𝑌_𝑡+^𝑡₀𝑓(𝑠)d𝑠+^𝑛_𝑗=1^𝑡₀𝑔_𝑗(𝑠)d𝛽_𝑗(𝑠)=𝑆_𝑡𝜂≥𝜂_𝑡=𝜉1_{𝑡=𝑇}+𝑆_𝑡1_{𝑡<𝑇}+^𝑡₀𝑓(𝑠)+d𝑠_{𝑛𝑗=1𝑡0}𝑔_𝑗(𝑠)d𝛽_𝑗(𝑠)≥𝑆_𝑇1_{𝑡=𝑇}+𝑆_𝑡1_{𝑡<𝑇}+^𝑡₀𝑓(𝑠)+d𝑠_{𝑛𝑗=1𝑡0}𝑔_𝑗(𝑠)d𝛽_𝑗(𝑠). (15) So, we have 𝑌_𝑡≥𝑆_𝑡.
Finally, from Hamadène [15], we get ∫^𝑇₀(𝑆_𝑡(𝜂)−𝜂_𝑡)d𝐾_𝑡=0; that is, ^𝑇₀𝑌_𝑡−𝑆_𝑡d𝐾_𝑡=0. (16) It shows that the process (𝑌_𝑡,𝑍_𝑡,𝐾_𝑡)_{0≤𝑡≤𝑇} is a solution of (4).

Theorem 3. Under the assumptions of (H1)–(H3), there exists a unique solution (𝑌_𝑡,𝑍_𝑡,𝐾_𝑡)_{0≤𝑡≤𝑇} of (3).

Proof. Let 𝒫=𝒮²×ℋ² be endowed with the norm ‖(𝑌,𝑍)‖_𝛽=𝐸^𝑇₀𝑒^𝛽𝑠𝑌_𝑠²+𝑍_𝑠²d𝑠^1/2 (17) for a suitable constant 𝛽>0. We define the map Φ from 𝒫 into itself and (𝑌,𝑍) and (𝑌^′,𝑍^′) are two elements of 𝒫. Define (𝑌,𝑍)=Φ(𝑌,𝑍), (𝑌^′,𝑍^′)=Φ(𝑌^′,𝑍^′), where (𝑌,𝑍,𝐾) and (𝑌^′,𝑍^′,𝐾^′) are solutions of (4) associated with (𝜉,𝑓(𝑡,𝑌,𝑍),𝑔_𝑗(𝑡,𝑌,𝑍),𝑆), and (𝜉,𝑓(𝑡,𝑌^′,𝑍^′),𝑔_𝑗(𝑡,𝑌^′,𝑍^′),𝑆^′), respectively. Set (𝑌,𝑍)=(𝑌_𝑡−𝑌^′_𝑡,𝑍_𝑡−𝑍^′_𝑡) and Ψ_𝑀(𝑥)=𝑥²𝟏_{{−𝑀≤𝑥≤𝑀}}+𝑀(2𝑥−𝑀)𝟏_{𝑥>𝑀}−𝑀(2𝑥+𝑀)𝟏_{{𝑥<−𝑀}}. (18) If we define Ψ^′_𝑀(𝑥)/𝑥=2, when 𝑥=0, then, 0≤Ψ^′_𝑀(𝑌_𝑠)/𝑌_𝑠≤2. Applying Itô formula to 𝑒^𝛽𝑡Ψ_𝑀(𝑌_𝑠), we have 𝑒^𝛽𝑡Ψ_𝑀𝑌_𝑡+𝛽^𝑇_𝑡𝑒^𝛽𝑠Ψ_𝑀𝑌_𝑠+d𝑠_𝑇𝑡𝑒^𝛽𝑠𝟏_{{−𝑀≤𝑌𝑠≤𝑀}}𝑍_𝑠²=d𝑠_𝑇𝑡𝑒^𝛽𝑠Ψ^′_𝑀𝑌_𝑠𝑓𝑠,𝑌_𝑠,𝑍_𝑠−𝑓𝑠,𝑌^′_𝑠,𝑍^′_𝑠+d𝑠_{𝑛𝑗=1𝑇𝑡}𝑒^𝛽𝑠𝟏_{{−𝑀≤𝑌𝑠≤𝑀}}𝑔_𝑗𝑠,𝑌_𝑠,𝑍_𝑠−𝑔_𝑗𝑠,𝑌^′_𝑠,𝑍^′_𝑠²−d𝑠_{𝑛𝑗=1𝑇𝑡}𝑒^𝛽𝑠Ψ^′_𝑀𝑌_𝑠𝑔_𝑗𝑠,𝑌_𝑠,𝑍_𝑠−𝑔_𝑗𝑠,𝑌^′_𝑠,𝑍^′_𝑠d𝛽_𝑗(𝑠)−^𝑇_𝑡𝑒^𝛽𝑠Ψ^′_𝑀𝑌_𝑠𝑍_𝑠(𝑠)+d𝑊_𝑇𝑡𝑒^𝛽𝑠Ψ^′_𝑀𝑌_𝑠d𝐾_𝑠−d𝐾^′_𝑠. (19) Taking expectation on both sides of (19) and noticing that ∫^𝑇_𝑡𝑒^𝛽𝑠Ψ^′_𝑀(𝑌_𝑠)(d𝐾_𝑠−d𝐾^′_𝑠)≤0, we have 𝐸𝑒^𝛽𝑡Ψ_𝑀𝑌_𝑡+𝐸𝛽^𝑇_𝑡𝑒^𝛽𝑠Ψ_𝑀𝑌_𝑠d𝑠+𝐸^𝑇_𝑡𝑒^𝛽𝑠𝟏_{{−𝑀≤𝑌𝑠≤𝑀}}𝑍_𝑠²d𝑠≤𝐸^𝑇_𝑡𝑒^𝛽𝑠Ψ^′_𝑀𝑌_𝑠𝑓𝑠,𝑌_𝑠,𝑍_𝑠−𝑓𝑠,𝑌^′_𝑠,𝑍^′_𝑠+d𝑠_𝑛𝑗=1𝐸^𝑇_𝑡𝑒^𝛽𝑠𝟏_{{−𝑀≤𝑌𝑠≤𝑀}}𝑔_𝑗𝑠,𝑌_𝑠,𝑍_𝑠−𝑔_𝑗𝑠,𝑌^′_𝑠,𝑍^′_𝑠²d𝑠≤2𝐸^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠𝑓𝑠,𝑌_𝑠,𝑍_𝑠−𝑓𝑠,𝑌^′_𝑠,𝑍^′_𝑠+d𝑠_𝑛𝑗=1𝐸^𝑇_𝑡𝑒^𝛽𝑠𝑔_𝑗𝑠,𝑌_𝑠,𝑍_𝑠−𝑔_𝑗𝑠,𝑌^′_𝑠,𝑍^′_𝑠²≤d𝑠2𝐶1−𝛼𝐸^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠²+d𝑠_∞𝑗=1𝐶_𝑗+1−𝛼2𝐸^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠−𝑌^′_𝑠²+d𝑠1+𝛼2𝐸^𝑇_𝑡𝑒^𝛽𝑠𝑍_𝑠−𝑍^′_𝑠²d𝑠. (20) Let 𝛾=2𝐶/(1−𝛼), 𝐶=2(∑^∞_𝑗=1𝐶_𝑗+((1−𝛼)/2))/(1+𝛼), 𝛽=𝛾+𝐶, and 𝑀→∞; we have 𝐶𝐸^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠−𝑌^′_𝑠²d𝑠+𝐸^𝑇_𝑡𝑒^𝛽𝑠𝑍_𝑠−𝑍^′_𝑠²≤d𝑠1+𝛼2𝐸^𝑇_𝑡𝑒^𝛽𝑠𝐶𝑌_𝑠−𝑌^′_𝑠²+𝑍_𝑠−𝑍^′_𝑠²; (21) that is, 𝑌_𝑠,𝑍_𝑠²_𝛽≤1+𝛼2𝑌^′_𝑠,𝑍^′_𝑠²_𝛽. (22)
It follows that Φ is a strict contraction on 𝒫 with the norm ‖⋅‖_𝛽, where 𝛽 is defined as above. Then, Φ has a fixed point (𝑌,𝑍,𝐾) which is the unique solution of (4) from the Burkholder-Davis-Gundy inequality.

With all the preparations, we will give the main result of this paper as follows.

Theorem 4. Under the conditions of (H1)–(H3), there exists a unique solution (𝑌_𝑡,𝑍_𝑡,𝐾_𝑡)_{0≤𝑡≤𝑇}∈𝒮²×ℋ²×𝒜² of (2).

Proof (existence). By Theorem 3, for any 𝑛≥1, there exists a unique solution of (3), denoted by (𝑌^𝑛_𝑡,𝑍^𝑛_𝑡,𝐾^𝑛_𝑡), 𝑌^𝑛_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝑌^𝑛_𝑠,𝑍^𝑛_𝑠+d𝑠_{𝑛𝑗=1𝑇𝑡}𝑔_𝑗𝑠,𝑌^𝑛_𝑠,𝑍^𝑛_𝑠d𝛽_𝑗(𝑠)−^𝑇_𝑡𝑍^𝑛_𝑠d𝑊(𝑠)+𝐾^𝑛_𝑇−𝐾^𝑛_𝑡. (23)
In the following parts, we will claim that (𝑌^𝑛_𝑡,𝑍^𝑛_𝑡,𝐾^𝑛_𝑡) is a Cauchy sequence in 𝒮²×ℋ²×𝒜². Without loss of generality, we let 𝑛<𝑚. Applying general Itô formula to |𝑌^𝑛_𝑡−𝑌^𝑚_𝑡|², we have. 𝑌^𝑛_𝑡−𝑌^𝑚_𝑡²+^𝑇_𝑡𝑍^𝑛_𝑠−𝑍^𝑚_𝑠²d𝑠=2^𝑇_𝑡𝑌^𝑛_𝑠−𝑌^𝑚_𝑠𝑓𝑠,𝑌^𝑛_𝑠,𝑍^𝑛_𝑠−𝑓𝑠,𝑌^𝑚_𝑠,𝑍^𝑚_𝑠+d𝑠_{𝑚𝑗=𝑛+1𝑇𝑡}𝑔_𝑗𝑠,𝑌^𝑛_𝑠,𝑍^𝑛_𝑠−𝑔_𝑗𝑠,𝑌^𝑚_𝑠,𝑍^𝑚_𝑠²d𝑠−2^𝑚_𝑗=𝑛+1^𝑇_𝑡𝑌^𝑛_𝑠−𝑌^𝑚_𝑠𝑔_𝑗𝑠,𝑌^𝑛_𝑠,𝑍^𝑛_𝑠−𝑔_𝑗𝑠,𝑌^𝑚_𝑠,𝑍^𝑚_𝑠d𝛽_𝑗(𝑠)−2^𝑇_𝑡𝑌^𝑛_𝑠−𝑌^𝑚_𝑠𝑍^𝑛_𝑠−𝑍^𝑚_𝑠d𝑊(𝑠)+2^𝑇_𝑡𝑌^𝑛_𝑠−𝑌^𝑚_𝑠d𝐾^𝑛_𝑠−d𝐾^𝑚_𝑠. (24) Taking expectation on both sides of (24) and noting that ∫^𝑇_𝑡(𝑌^𝑛_𝑠−𝑌^𝑚_𝑠)(d𝐾^𝑛_𝑠−d𝐾^𝑚_𝑠)≤0, we obtain 𝐸𝑌^𝑛_𝑡−𝑌^𝑚_𝑡²+𝐸^𝑇_𝑡𝑍^𝑛_𝑠−𝑍^𝑚_𝑠²d𝑠≤2𝐸^𝑇_𝑡𝑌^𝑛_𝑠−𝑌^𝑚_𝑠𝑓𝑠,𝑌^𝑛_𝑠,𝑍^𝑛_𝑠−𝑓𝑠,𝑌^𝑚_𝑠,𝑍^𝑚_𝑠+d𝑠_{𝑚𝑗=𝑛+1}𝐸^𝑇_𝑡𝑔_𝑗𝑠,𝑌^𝑛_𝑠,𝑍^𝑛_𝑠−𝑔_𝑗𝑠,𝑌^𝑚_𝑠,𝑍^𝑚_𝑠²d𝑠. (25) By (H3) and elementary inequality 2𝑎𝑏≤𝛽𝑎²+(1/𝛽)𝑏², 𝛽>0, we obtain 𝐸𝑌^𝑛_𝑡−𝑌^𝑚_𝑡²+𝐸^𝑇_𝑡𝑍^𝑛_𝑠−𝑍^𝑚_𝑠²≤d𝑠2𝐶𝐸1−𝛼^𝑇_𝑡𝑌^𝑛_𝑠−𝑌^𝑚_𝑠²d𝑠+1−𝛼2𝐸^𝑇_𝑡𝑌^𝑛_𝑠−𝑌^𝑚_𝑠²+d𝑠1−𝛼2𝐸^𝑇_𝑡𝑍^𝑛_𝑠−𝑍^𝑚_𝑠²d𝑠+𝛼𝐸^𝑇_𝑡𝑍^𝑛_𝑠−𝑍^𝑚_𝑠²+d𝑠_{𝑚𝑗=𝑛+1}𝐶_𝑗𝐸^𝑇_𝑡𝑌^𝑛_𝑠−𝑌^𝑚_𝑠²d𝑠. (26) Furthermore, 𝐸𝑌^𝑛_𝑡−𝑌^𝑚_𝑡²+1−𝛼2𝐸^𝑇_𝑡𝑍^𝑛_𝑠−𝑍^𝑚_𝑠²d𝑠≤𝐶_𝑝𝐸^𝑇_𝑡𝑌^𝑛_𝑠−𝑌^𝑚_𝑠²d𝑠, (27) where 𝐶_𝑝=(2𝐶/(1−𝛼))+((1−𝛼)/2)+∑^𝑚_𝑗=𝑛+1𝐶_𝑗.
By Gronwall’s inequality and Burkholder-Davis-Gundy inequality, we have 𝐸sup_{0≤𝑡≤𝑇}^𝑇_𝑡𝑌^𝑛_𝑠−𝑌^𝑚_𝑠²d𝑠⟶0. (28)
Denote the limit of (𝑌^𝑛_𝑡,𝑍^𝑛_𝑡,𝐾^𝑛_𝑡) by (𝑌_𝑡,𝑍_𝑡,𝐾_𝑡); we will show that (𝑌_𝑡,𝑍_𝑡,𝐾_𝑡) satisfies (2). If it is necessary, we can choose a subsequence of (3). By Hölder’s inequality, 𝐸^𝑇_𝑡𝑓𝑠,𝑌_𝑠,𝑍_𝑠−𝑓𝑠,𝑌^𝑛_𝑠,𝑍^𝑛_𝑠d𝑠²≤𝑇𝐸^𝑇_𝑡𝑓𝑠,𝑌_𝑠,𝑍_𝑠−𝑓𝑠,𝑌^𝑛_𝑠,𝑍^𝑛_𝑠²d𝑠⟶0. (29) From (27), we know 𝐸^𝑇₀𝑌^𝑛_𝑡−𝑌_𝑡²d𝑡⟶0, (30) and 𝑌^𝑛_𝑡→𝑌_𝑡, a.e., so 𝐸^𝑇₀𝑌_𝑡^𝑛+1−𝑌^𝑛_𝑡²1d𝑡≤2^𝑛. (31) For any 𝑛, 𝑌^𝑛_𝑡≤𝑌¹_𝑡+^𝑛−1_𝑖=1𝑌_𝑡^𝑖+1−𝑌^𝑖_𝑡≤𝑌¹_𝑡+^∞_𝑖=1𝑌_𝑡^𝑖+1−𝑌^𝑖_𝑡. (32) Then, we have 𝐸^𝑇₀sup_𝑛𝑌^𝑛_𝑡²≤d𝑡𝐸^𝑇₀𝑌¹_𝑡+^∞_𝑖=1𝑌_𝑡^𝑖+1−𝑌^𝑖_𝑡²≤d𝑡𝐸^𝑇₀𝑌¹_𝑡²d𝑡+^∞_𝑖=1𝐸^𝑇₀𝑌_𝑡^𝑖+1−𝑌^𝑖_𝑡²≤d𝑡𝐸^𝑇₀𝑌¹_𝑡²d𝑡+^∞_𝑖=112^𝑖. (33) From (H4), it follows 𝐸^𝑇₀sup_𝑛𝑓𝑠,𝑌_𝑠,𝑍_𝑠−𝑓𝑠,𝑌^𝑛_𝑠,𝑍^𝑛_𝑠²d𝑠≤2𝐶𝐸^𝑇₀sup_𝑛𝑌^𝑛_𝑠²+𝑌_𝑠²+sup_𝑛𝑍^𝑛_𝑠²+𝑍_𝑠²d𝑠<∞. (34) Applying Lebesgue dominated convergence theorem, we deduce that (𝑌_𝑡,𝑍_𝑡,𝐾_𝑡) is the solution of (2) by continuity of the functions 𝑓 and 𝑔.
Uniqueness. Let (𝑌^𝑖_𝑡,𝑍^𝑖_𝑡,𝐾^𝑖_𝑡) (𝑖=1,2) be two solutions of (2), 𝑌_𝑡=𝑌¹_𝑡−𝑌²_𝑡, 𝑍_𝑡=𝑍¹_𝑡−𝑍²_𝑡. We apply Itô formula to 𝑒^𝛽𝑡Ψ_𝑀(𝑌_𝑡), for any 𝛽∈ℝ, 𝑒^𝛽𝑡Ψ_𝑀𝑌_𝑡+𝛽^𝑇_𝑡𝑒^𝛽𝑠Ψ_𝑀𝑌_𝑠+d𝑠_𝑇𝑡𝑒^𝛽𝑠𝟏_{{−𝑀≤𝑌𝑠≤𝑀}}𝑍_𝑠²=d𝑠_𝑇𝑡𝑒^𝛽𝑠Ψ^′_𝑀𝑌_𝑠𝑓𝑠,𝑌¹_𝑠,𝑍¹_𝑠−𝑓𝑠,𝑌²_𝑠,𝑍²_𝑠+d𝑠_{∞𝑗=1𝑇𝑡}𝑒^𝛽𝑠𝟏_{{−𝑀≤𝑌𝑠≤𝑀}}𝑔_𝑗𝑠,𝑌¹_𝑠,𝑍¹_𝑠−𝑔_𝑗𝑠,𝑌²_𝑠,𝑍²_𝑠²−d𝑠_{∞𝑗=1𝑇𝑡}𝑒^𝛽𝑠Ψ^′_𝑀𝑌_𝑠×𝑔_𝑗𝑠,𝑌¹_𝑠,𝑍¹_𝑠−𝑔_𝑗𝑠,𝑌²_𝑠,𝑍²_𝑠d𝛽_𝑗(𝑠)−^𝑇_𝑡𝑒^𝛽𝑠Ψ^′_𝑀𝑌_𝑠𝑍_𝑠d𝑊_𝑠+^𝑇_𝑡𝑒^𝛽𝑠Ψ^′_𝑀𝑌_𝑠d𝐾¹_𝑠−d𝐾²_𝑠. (35) Taking expectation on both sides of (35), 𝐸𝑒^𝛽𝑡Ψ_𝑀𝑌_𝑡+𝛽𝐸^𝑇_𝑡𝑒^𝛽𝑠Ψ_𝑀𝑌_𝑠d𝑠+𝐸^𝑇_𝑡𝑒^𝛽𝑠𝟏_{{−𝑀≤𝑌𝑠≤𝑀}}𝑍_𝑠²d𝑠≤2𝐸^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠𝑓𝑠,𝑌¹_𝑠,𝑍¹_𝑠−𝑓𝑠,𝑌²_𝑠,𝑍²_𝑠+d𝑠_∞𝑗=1𝐸^𝑇_𝑡𝑒^𝛽𝑠𝟏_{{−𝑀≤𝑌𝑠≤𝑀}}𝑔_𝑗𝑠,𝑌¹_𝑠,𝑍¹_𝑠−𝑔_𝑗𝑠,𝑌²_𝑠,𝑍²_𝑠²≤d𝑠2𝐶1−∑^∞_𝑗=1𝛼_𝑗+^∞_𝑗=1𝐶_𝑗+1−∑^∞_𝑗=1𝛼_𝑗2𝐸^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠²+d𝑠1+∑^∞_𝑗=1𝛼_𝑗2𝐸^𝑇_𝑡𝑒^𝛽𝑠𝑍_𝑡²d𝑠. (36) Let 𝑀→∞, and applying monotone convergence theorem, we have 𝐸𝑒^𝛽𝑡𝑌_𝑡²+𝛽−2𝐶−1−𝛼^∞_𝑗=1𝐶_𝑗−1−𝛼2×𝐸^𝑇_𝑡𝑒^𝛽𝑠𝑌_𝑠²+d𝑠1−𝛼2𝐸^𝑇_𝑡𝑒^𝛽𝑠𝑍_𝑠²d𝑠≤0. (37)
When 𝛽 is taken sufficiently large, we have 𝑌_𝑡=0, a.e., for all 𝑠∈[𝑡,𝑇]. So, we have 𝑍_𝑡=0, a.e. Then, we complete the proof.

Acknowledgments

The authors would like to take this chance to express their sincere gratitude to the National Natural Science Foundation of China (11201004 and 11371029), Natural Science Foundation of Anhui Province (KJ2011B176 and KJ2013B288), Professors(Doctors) Scientific Research Foundation of Suzhou University (2013jb04), and Foundation of Laboratory of Intelligent Information Processing of Suzhou University (2010YKF11).