Nonlinear Decomposition of Doob-Meyer's Type for Continuous 𝑔-Supermartingale with Uniformly Continuous Coefficient

Abstract

We prove that a continuous 𝑔-supermartingale with uniformly continuous coeffcient 𝑔 on finite or infinite horizon, is a 𝑔-supersolution of the corresponding backward stochastic differential equation. It is a new nonlinear Doob-Meyer decomposition theorem for the 𝑔-supermartingale with continuous trajectory.

1. Introduction

In 1990, Pardoux-Peng [1] proposed the following nonlinear backward stochastic differential equation (BSDE) driven by a Brownian motion: 𝑦_𝑡=𝜉+^𝑇_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠−^𝑇_𝑡𝑧_𝑠𝑑𝐵_𝑠,𝑡∈[0,𝑇], (1) where the positive real number𝑇, the random variable𝜉, and the function𝑔are called the time horizon, the terminal data, and the generator, respectively, and the pair of adapted processes(𝑦_𝑡,𝑧_𝑡)_{𝑡∈[0,𝑇]}to be known is called the solution of the BSDE (1). In this paper, we study a more generalized BSDE with a given increasing process(𝑉_𝑡)_{𝑡∈[0,𝑇]}with𝑉₀=0: 𝑦_𝑡=𝜉+^𝑇_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠−^𝑇_𝑡𝑧_𝑠𝑑𝐵_𝑠+𝑉_𝑇−𝑉_𝑡,𝑡∈[0,𝑇]. (2) If(𝑉_𝑡)_{𝑡∈[0,𝑇]}≡0, the first component(𝑦_𝑡)_{𝑡∈[0,𝑇]}of solution of (2) is called the𝑔-solution of (1); otherwise, it is called the𝑔-supersolution. Subsequently, Peng [2] introduced the nonlinear expectation and nonlinear martingale theories via BSDEs. In [3], Peng first obtained the monotonic limit theorem; that is, under some mild conditions, the limit of a monotonically increasing sequence of𝑔-supersolutions is also a𝑔-supersolution. And applying this result, he proved that a càdlàg𝑔-martingale, which is right continuous with left limits, had a nonlinear decomposition of Doob-Meyer’s type, corresponding to the classical martingale theory. Later, Lin [4, 5] extended Peng’s result and got this decomposition for the𝑔-supermartingale with respect to a general continuous filtration and that with jumps, respectively. It should be pointed out that, in Peng [3] and Lin [4, 5], the monotonic limit theorem for BSDEs plays a key role, and it is also useful in other problems. For example, in [6], Peng-Xu put forward a generalized version of monotonic limit theorem and proved that solving the reflected BSDE with a given lower barrier process was equivalent to finding the smallest𝑔-supermartingale dominating the barrier. And Peng-Xu [7] used this technique to treat the problems of the BSDE with generalized constraints and solve the American option pricing problem in an incomplete market. On the other hand, motivated by the theories of the classical martingale and the nonlinear martingale, Chen-Wang [8] showed that the BSDEs on infinite time horizon were solvable, under the Lipschitz assumption on𝑔, whose Lipschitzian coefficient is a function depending on𝑡, and they obtained the convergence theorem of the nonlinear𝑔-martingale. Afterward, Fan et al. [9] explored the BSDEs on finite or infinite horizon, without the Lipschitz assumption, and got an existence and uniqueness result and a comparison theorem.

Based on these results, a natural question is, under the generalized uniformly continuous assumption on the coefficient𝑔, does the𝑔-martingale still have a nonlinear decomposition of Doob-Meyer’s type? Our answer is yes. We prove that if a𝑔-supermartingale has a continuous trajectory on finite or infinite time interval, then it is a𝑔-supersolution of the corresponding BSDE; that is, it has a nonlinear Doob-Meyer decomposition. It should be noted that our results are based on the conditions without the Lipschitz assumption on the coefficient𝑔. And our results do not depend on the infinite time version of the monotonic limit but only on the penalization method.

The outline of this paper is as follows. Section 2 provides some assumptions, definitions, and the existence and uniqueness theorem and comparison theorem for a generalized BSDE with generalized uniformly continuous generator𝑔. Then, Section 3 devotes to the main result a new version of nonlinear Doob-Meyer’s decomposition theorem for the continuous𝑔-supermartingale with the generalized uniformly continuous coefficient.

2. Preliminaries

Let𝑇be a finite or infinite nonnegative extended real number, and let(𝐵_𝑡)_𝑡≥0be a standard𝑑-dimensional Brownian motion defined on a complete probability space(Ω,ℱ,𝑃)endowed with a filtration(ℱ_𝑡)_𝑡≥0generated by this Brownian motion: ℱ_𝑡≜𝜎𝐵_𝑠∶0≤𝑠≤𝑡∨𝒩,ℱ_∞=_𝑡≥0ℱ_𝑡, (3) where 𝒩 is the set of all𝑃-null subsets.

For simplicity of presentation, we use|𝑥|to denote the Euclidean norm of𝑥in ℝ or ℝ^𝑑, and let𝐿²(Ω,ℱ_𝑡,𝑃)be the space of all theℱ_𝑡measurable square integrable real valued random variables, and define the adapted process spaces as follows: ℋ²(0,𝜏;ℝ^𝑑)∶={(𝜙_𝑡)_{𝑡∈[0,𝜏]} is a ℝ^𝑑-valued process such that 𝔼[∫^𝜏₀|𝜙_𝑡|²𝑑𝑡]<+∞}; 𝒮²(0,𝜏;ℝ)∶={(𝜙_𝑡)_{𝑡∈[0,𝜏]} is a càdlàg ℝ- valued process such that 𝔼[sup_{0≤𝑡≤𝜏}|𝜙_𝑡|²]<+∞}; 𝒜²(0,𝜏;ℝ)∶={(𝜙_𝑡)_{𝑡∈[0,𝜏]} is an increasing process in 𝒮²(0,𝜏;ℝ) with 𝜙(0)=0}.Clearly, all the above spaces of stochastic processes are completed Banach spaces.

Furthermore, we denote the set of linear increasing functions𝜙(⋅)∶ℝ₊↦ℝ₊with𝜙(0)=0by 𝒳. Here the linear increasing means that, for any element𝜙∈𝒳, there exists a pair of positive real numbers(𝑎,𝑏)depending on𝜙such that, for all 𝑥∈ℝ₊, 𝜙(𝑥)≤𝑎𝑥+𝑏.

The generator𝑔(𝑡,𝜔,𝑦,𝑧)∶[0,𝑇]×Ω×ℝ×ℝ^𝑑↦ℝ is a random function which is a progressively measurable stochastic process for any(𝑦,𝑧). We assume that it satisfies the following two assumptions, where (H2) is a generalized uniformly continuous condition; that is, its modulus of continuity may depend on𝑡:(H1)𝔼[(∫^𝑇₀|𝑔(𝑡,0,0)|𝑑𝑡)²]<∞;(H2)|𝑔(𝑡,𝑦,𝑧)−𝑔(𝑡,𝑦^′,𝑧^′)|≤𝑢(𝑡)𝜑(|𝑦−𝑦^′|)+𝑣(𝑡)𝜓(|𝑧−𝑧^′|), where𝑢(⋅)and𝑣(⋅)are two positive functions mapping fromℝ⁺toℝ⁺, such that∫^𝑇₀[𝑢(𝑡)+𝑣²(𝑡)]𝑑𝑡<∞; the functions𝜑and𝜓belong to𝒳 and 𝜑(⋅)is a concave function, with∫₀₊𝜑(𝑡)𝑑𝑡=+∞. And in addition, we assume that∫^𝑇₀𝑣(𝑡)𝑑𝑡<∞, if𝜓cannot be dominated by a linear function; that is, we cannot find a real number𝑎, such that𝜙(𝑥)≤𝑎𝑥.

Remark 1. In (H2), 𝜑(⋅)is a concave function which means that𝜑(𝜆𝑡₁+(1−𝜆)𝑡₂)≥𝜆𝜑(𝑡₁)+(1−𝜆)𝜑(𝑡₂), for𝜆∈[0,1]and𝑡₁,𝑡₂∈[0,𝑇]. And the equality∫₀₊𝜑(𝑡)𝑑𝑡=∞means that the value of the integration∫^𝛿₀𝜑(𝑡)𝑑𝑡will be infinite on any interval[0,𝛿]with𝛿>0. For simplicity, we also use𝑢_𝑠and𝑣_𝑠to denote𝑢(𝑠)and𝑣(𝑠), respectively, in the remaining of this paper.

Now, we consider the following problem. Suppose that the time horizon𝑇, generator𝑔, terminal data𝑦_𝑇, and the increasing càdlàg process(𝑉_𝑡)_{𝑡∈[0,𝑇]}∈𝒮²(0,𝑇;ℝ) are given in advance; let us find a pair of processes(𝑦_𝑡,𝑧_𝑡)_{𝑡∈[0,𝑇]}∈𝒮²(0,𝑇;ℝ)×ℋ²(0,𝑇;ℝ^𝑑)satisfying 𝑦_𝑡=𝑦_𝑇+^𝑇_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠−^𝑇_𝑡𝑧_𝑠𝑑𝐵_𝑠+𝑉_𝑇−𝑉_𝑡,𝑡∈[0,𝑇]. (4)

If 𝑉_𝑡≡0, the above equation (4) will be a classical BSDE on finite or infinite horizon; the existence and uniqueness result is already obtained, which is stated by Theorem 3 in Fan et al. [9]. Otherwise we can set𝑦_𝑡∶=𝑦_𝑡+𝑉_𝑡and treat the following BSDE as 𝑦_𝑡=𝑦_𝑇+𝑉_𝑇+^𝑇_𝑡𝑔𝑠,𝑦_𝑠−𝑉_𝑠,𝑧_𝑠𝑑𝑠−^𝑇_𝑡𝑧_𝑠𝑑𝐵_𝑠. (5) It is a classical BSDE with the terminal data𝜉∶=𝑦_𝑇+𝑉_𝑇and the generator𝑔∶=𝑔(𝑡,𝑦−𝑉_𝑡,𝑧). Since the assumptions (H1) and (H2) hold for generator𝑔, it is easy to verify that𝑔still satisfies the two conditions. So we have the following existence and uniqueness theorem.

Lemma 2 (existence and uniqueness). One assumes that the generator𝑔of the BSDE (4) satisfies the conditions (H1) and (H2). Then, for any random variable𝑦_𝑇∈𝐿²(Ω,ℱ_𝑇,𝑃), and a process(𝑉_𝑡)_{𝑡∈[0,𝑇]}∈𝒮²(0,𝑇;ℝ), there exists a unique pair of processes(𝑦_𝑡,𝑧_𝑡)_{𝑡∈[0,𝑇]}∈𝒮²(0,𝑇;ℝ)×ℋ²(0,𝑇;ℝ^𝑑), which is a solution of the BSDE (4), such that(𝑦_𝑡+𝑉_𝑡)_{𝑡∈[0,𝑇]}is continuous and 𝔼sup_{0≤𝑡≤𝑇}𝑦_𝑡²<∞. (6)

We can also have the following comparison theorem, which will be used in the latter part of this section and the next one.

Proposition 3 (comparison). Suppose that the assumptions in Lemma 2 hold. Let(𝑦_𝑡,𝑧_𝑡)_{𝑡∈[0,𝑇]}be the solution of another BSDE: 𝑦_𝑡=𝑦_𝑇+^𝑇_𝑡𝑔_𝑠𝑑𝑠−^𝑇_𝑡𝑧_𝑠𝑑𝐵_𝑠+𝑉_𝑇−𝑉_𝑡,𝑡∈[0,𝑇], (7) where(𝑉_𝑡)_{𝑡∈[0,𝑇]}∈𝒮²(0,𝑇;ℝ),𝑦_𝑇∈𝐿²(Ω,ℱ_𝑇,𝑃), and (𝑔_𝑡)_{𝑡∈[0,𝑇]}are given such that(1)𝑦_𝑇∶=𝑦_𝑇−𝑦_𝑇≥0; (2)𝑔_𝑡∶=𝑔(𝑡,𝑦_𝑡,𝑧_𝑡)−𝑔_𝑡≥0, 𝑑𝑃×𝑑𝑡-a.e.;(3)𝑉_𝑡∶=𝑉_𝑡−𝑉_𝑡 is a càdlàg increasing process;(4)𝔼[(∫^𝑇₀|𝑔_𝑡|𝑑𝑡)²]<∞.Then we have, P-a.s., 𝑦_𝑡≥𝑦_𝑡,∀𝑡∈[0,𝑇]. (8)

Proof. We sketch the proof as follows. Set𝑦_𝑡=𝑦_𝑡−𝑦_𝑡 and 𝑧_𝑡=𝑧_𝑡−𝑧_𝑡; applying Itô-Meyer’s formula to(𝑦_𝑡−𝑦_𝑡)⁺leads to 𝑦⁺_𝑡≤𝑦_𝑇−𝑦_𝑇⁺+^𝑇_𝑡1_{𝑦_𝑠^>0}𝑔_𝑠−𝑔𝑠,𝑦_𝑠,𝑧_𝑠−𝑑𝑠_𝑇𝑡1_{𝑦_𝑠^>0}𝑧_𝑠𝑑𝐵_𝑠+^𝑇_𝑡1_{𝑦_𝑠−^>0}𝑑𝑉_𝑠−𝑑𝑉_𝑠. (9) Since𝑉_𝑡is an increasing process, we see that ^𝑇_𝑡1_{𝑦_𝑠−^>0}𝑑𝑉_𝑠−𝑑𝑉_𝑠≤−^𝑇_𝑡1_{𝑦_𝑠−^>0}𝑑𝑉_𝑠≤0. (10)
Recalling that𝑔_𝑡∶=𝑔(𝑡,𝑦_𝑡,𝑧_𝑡)−𝑔_𝑡≥0,𝑑𝑃×𝑑𝑡-a.e., and the assumption (H2), we can get 1_{𝑦_𝑠>0}𝑔_𝑠−𝑔𝑠,𝑦_𝑠,𝑧_𝑠=1_{𝑦_𝑠^>0}𝑔_𝑠−𝑔𝑠,𝑦_𝑠,𝑧_𝑠+𝑔𝑠,𝑦_𝑠,𝑧_𝑠−𝑔𝑠,𝑦_𝑠,𝑧_𝑠≤𝑢_𝑠𝜑𝑦⁺_𝑠+1_{𝑦_𝑠>0}𝑣_𝑠𝜓𝑧_𝑠. (11) Thus, it follows that 𝑦⁺_𝑡≤^𝑇_𝑡𝑢_𝑠𝜑𝑦⁺_𝑠+1_{𝑦_𝑠^>0}𝑣_𝑠𝜓𝑧_𝑠𝑑𝑠−^𝑇_𝑡1_{𝑦_𝑠^>0}𝑧_𝑠𝑑𝐵_𝑠. (12) Now we are in the same position with Theorem 2 in Fan et al. [9]. Then we can prove that, for all 𝑡∈[0,𝑇],𝑦⁺_𝑡≤0, P-a.s. Therefore, for any𝑡∈[0,𝑇], we have 𝑦_𝑡≤𝑦_𝑡,P-a.s. (13) Observing that𝑦_𝑡and𝑦_𝑡are càdlàg processes, we can conclude that, P-a.s., 𝑦_𝑡≤𝑦_𝑡,∀𝑡∈[0,𝑇]. (14)

Remark 4. If we replace the deterministic terminal time𝑇by aℱ_𝑡-stopping time𝜏≤𝑇, then, by Lemma 2, existence and uniqueness theorem and the above comparison theorem still hold true.

For a given stopping time𝜏≤𝑇, we now consider the following BSDE: 𝑦_𝑡=𝑦_𝜏+^𝜏_𝑡∧𝜏𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠−^𝜏_𝑡∧𝜏𝑧_𝑠𝑑𝐵_𝑠+𝐴_𝜏−𝐴_𝑡∧𝜏, (15) where𝑦_𝜏∈𝐿²(Ω,ℱ_𝜏,𝑃) and (𝐴_𝑡)_{𝑡∈[0,𝑇]}∈𝒜²(0,𝜏;ℝ) is a given càdlàg increasing process with𝐴₀=0.

Next, we introduce the conceptions of𝑔-solution,𝑔-supersolution,𝑔-martingale, and 𝑔-supermartingale closely following Peng’s definitions in [3].

Definition 5. If a process(𝑦_𝑡)_{𝑡∈[0,𝜏]}can be written in the form of the BSDE (15) with the generator𝑔, then one call it a g-supersolution on[0,𝜏]. Particularly, if(𝐴_𝑡)_{𝑡∈[0,𝜏]}≡0on[0,𝜏], then one call(𝑦_𝑡)_{𝑡∈[0,𝜏]}a𝑔-solution on[0,𝜏].

Definition 6. Aℱ_𝑡-progressively measurable real-valued process(𝑌_𝑡)_{𝑡∈[0,𝜏]}is called a𝑔-supermartingale (resp.𝑔-martingale), if for each stopping time𝜏≤𝑇, 𝔼|𝑌_𝜏|²<∞, and the𝑔-solution(𝑦_𝑡)_{𝑡∈[0,𝜏]}with terminal condition𝑦_𝜏=𝑌_𝜏satisfies𝑦_𝜎≤𝑌_𝜎(resp.𝑦_𝜎=𝑌_𝜎) for all stopping time𝜎≤𝜏. Indeed, a𝑔-martingale on[0,𝑇]is a𝑔-solution on[0,𝑇].

From Proposition 3, we know that a 𝑔-supersolution is a𝑔-supermartingale. Conversely, a meaningful and interesting question follows immediately. Is a𝑔-supermartingale a𝑔-supersolution? If so, does the𝑔-supermartingale, or𝑔-supersolution, has a unique representation of the form (15)?

According to Proposition  1.6 in [3], we can assert that, given a𝑔-supersolution(𝑦_𝑡)_{𝑡∈[0,𝜏]}on[0,𝜏], there is a unique pair of processes(𝑧_𝑡,𝐴_𝑡)_{𝑡∈[0,𝜏]}∈ℋ²(0,𝜏;ℝ^𝑑)×𝒜²(0,𝜏;ℝ) on[0,𝜏]such that the triple(𝑦_𝑡,𝑧_𝑡,𝐴_𝑡)_{𝑡∈[0,𝜏]}satisfies the BSDE (15). Now, we can propose the next conception as follows.

Definition 7. Provided that the process(𝑦_𝑡)_{𝑡∈[0,𝑇]}is a𝑔-supersolution and the triple of processes(𝑦_𝑡,𝑧_𝑡,𝐴_𝑡)_{𝑡∈[0,𝜏]}satisfies the BSDE (15), one call(𝑧_𝑡,𝐴_𝑡)_{𝑡∈[0,𝜏]}the unique decomposition of(𝑦_𝑡)_{𝑡∈[0,𝜏]}.

3. Nonlinear Doob-Meyer’s Decomposition for 𝑔-Supermartingale with Uniformly Continuous Coefficient

In this section, we provide and prove the main result of this paper that a continuous𝑔-supermartingale is a𝑔-supersolution; that is, it has a unique decomposition in the sense of Definition 7.

Theorem 8. One assumes that𝑔satisfies the conditions(𝐻1)and(𝐻2). Let(𝑌_𝑡)_{𝑡∈[0,𝑇]}be a continuous𝑔-supermartingale on[0,𝑇]in𝒮²(0,𝑇;ℝ). Then(𝑌_𝑡)_{𝑡∈[0,𝑇]}is a𝑔-supersolution on[0,𝑇] that is, there is a unique pair of processes(𝑧_𝑡,𝐴_𝑡)_{𝑡∈[0,𝑇]}inℋ²(0,𝑇;ℝ^𝑑)×𝒜²(0,𝑇;ℝ), such that(𝑌_𝑡)_{𝑡∈[0,𝑇]}coincides with the first component(𝑦_𝑡)_{𝑡∈[0,𝑇]}of the solution for the following BSDE: 𝑦_𝑡=𝑌_𝑇+^𝑇_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠−^𝑇_𝑡𝑧_𝑠𝑑𝐵_𝑠+𝐴_𝑇−𝐴_𝑡,𝑡∈[0,𝑇]. (16)

In order to prove this theorem, we consider the family of penalization BSDEs parameterized by𝑛=1,2,3,…, 𝑦^𝑛_𝑡=𝑌_𝑇+^𝑇_𝑡𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠𝑑𝑠−^𝑇_𝑡𝑧^𝑛_𝑠𝑑𝐵_𝑠+𝑛^𝑇_𝑡𝑣²_𝑠𝑌_𝑠−𝑦^𝑛_𝑠𝑑𝑠,𝑡∈[0,𝑇], (17) and set 𝐴^𝑛_𝑡∶=𝑛^𝑡₀𝑣²_𝑠𝑌_𝑠−𝑦^𝑛_𝑠𝑑𝑠,𝑡∈[0,𝑇]. (18) We first claim the next proposition.

Proposition 9. For each 𝑛=1,2,…, one has, P-a.s., 𝑌_𝑡≥𝑦^𝑛_𝑡,𝑡∈[0,𝑇]. (19)

Proof. Using an argument similar to that in Lemma  3.4 in [3], one can carry out the proof by contradiction. We sketch it as follows.
Supposing that it is not the case, then there exist𝛿>0and a positive integer𝑛such that the measure of{(𝜔,𝑡)∣𝑦^𝑛_𝑡−𝑌_𝑡−𝛿≥0}⊂Ω×[0,𝑇]is nonzero; then we can define the following stopping times: 𝜎∶=min𝑇,inf𝑡∣𝑦^𝑛_𝑡≥𝑌_𝑡+𝛿,𝜏∶=inf𝑡≥𝜎∣𝑦^𝑛_𝑡≤𝑌_𝑡. (20) It is observed, from the above definition and the continuous of(𝑌_𝑡)_{𝑡∈[0,𝑇]}, that𝜎≤𝜏≤𝑇 and 𝑃(𝜏>𝜎)>0. And furthermore, we have, P-a.s., (i)𝑦^𝑛_𝜎≥𝑌_𝜎(+𝛿;ii)𝑦^𝑛_𝜏≤𝑌_𝜏. (21) Now let(𝑦_𝑡)_{𝑡∈[0,𝜏]}(resp.(𝑦^′_𝑡)_{𝑡∈[0,𝜏]}) be the𝑔-solution on[0,𝜏]with terminal condition𝑦_𝜏=𝑦^𝑛_𝜏(resp.𝑦^′_𝜏=𝑌_𝜏). By Proposition 3, (21)-(ii) implies that𝑦^𝑛_𝜎≤𝑦_𝜎≤𝑦^′_𝜎. On the other hand since(𝑌_𝑡)_{𝑡∈[0,𝑇]}is a𝑔-supermartingale, thus we can get 𝑌_𝜎≥𝑦^𝑛_𝜎,P-a.s. (22) This is a contradiction to (21)-(i). Then by Fubini’s theorem, we have, P-a.s., 𝑌_𝑡≥𝑦^𝑛_𝑡,dt-a.e. (23) And the conclusion follows from the continuity of(𝑌_𝑡)_{𝑡∈[0,𝑇]}(𝑦^𝑛_𝑡)_{𝑡∈[0,𝑇]}. The proof is completed.

Now, we can get the following result; the boundedness of the triple of the processes(𝑦^𝑛_𝑡,𝑧^𝑛_𝑡,𝐴^𝑛_𝑡)_{𝑡∈[0,𝑇]}can be defined by the penalization BSDEs.

Proposition 10. There exists a positive real number C such that for any positive integer𝑛𝔼sup_{0≤𝑡≤𝑇}𝑦^𝑛_𝑡²+^𝑇₀𝑧^𝑛_𝑡²𝑑𝑡+𝐴^𝑛_𝑇²≤𝐶. (24)

Proof. From BSDE (17), we have 𝐴^𝑛_𝑇=𝑦^𝑛₀−𝑌_𝑇−^𝑇₀𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠𝑑𝑠+^𝑇₀𝑧^𝑛_𝑠𝑑𝐵_𝑠≤𝑦^𝑛₀+𝑌_𝑇+^𝑇₀𝑔(𝑠,0,0)+𝑢_𝑠𝜑𝑦^𝑛_𝑠+𝑣_𝑠𝜓𝑧^𝑛_s𝑑𝑠+^𝑇₀𝑧^𝑛_𝑠𝑑𝐵_𝑠≤𝑦^𝑛₀+𝑌_𝑇+^𝑇₀𝑔(𝑠,0,0)+𝑢_𝑠𝑎_𝜑𝑦^𝑛_𝑠+𝑏_𝜑+𝑣_𝑠𝑎_𝜓𝑧^𝑛_𝑠+𝑏_𝜙𝑑𝑠+^𝑇₀𝑧^𝑛_𝑠𝑑𝐵_𝑠, (25) where the real numbers𝑎_𝜑,𝑏_𝜑 and 𝑎_𝜓,𝑏_𝜓depend on the functions𝜑 and 𝜓, respectively. From Proposition 9, we see that𝑦^𝑛_𝑡is dominated by|𝑦¹_𝑡|+|𝑌_𝑡|. Thus there exists a constant𝐶₀independent of𝑛, such that 𝔼sup_{0≤𝑡≤𝑇}|𝑦^𝑛_𝑡|²≤𝐶₀. (26) Now, noticing the boundedness of(𝑦^𝑛_𝑡)_{𝑡∈[0,𝑇]}in the above sense, from the basic algebraic inequality, Jensen’s inequality and Hölder’s inequality, we can get that there exists another constant𝐶₁such that 𝔼𝐴^𝑛_𝑇²≤8𝔼𝑦^𝑛₀²+𝑌_𝑇²+^𝑇₀𝑔(𝑠,0,0)𝑑𝑠²+𝑎²_𝜑^𝑇₀𝑢_𝑠𝑑𝑠²sup_{0≤𝑡≤𝑇}𝑦^𝑛_𝑡²+𝑏²_𝜑^𝑇₀𝑢_𝑠𝑑𝑠²+𝑎²_𝜓^𝑇₀𝑣²_𝑠𝑑𝑠^𝑇₀𝑧^𝑛_𝑠²𝑑𝑠+𝑏²_𝜓^𝑇₀𝑣_𝑠𝑑𝑠²+^𝑇₀𝑧^𝑛_𝑠²𝑑𝑠≤𝐶₁+𝐶₁𝔼^𝑇₀𝑧^𝑛_𝑠²𝑑𝑠. (27)
On the other hand, in the light of (H2), applying Itô’s formula to|𝑦^𝑛_𝑡|²on[0,𝑇]will lead to 𝑦^𝑛₀²+𝔼^𝑇₀𝑧^𝑛_𝑠²𝑌𝑑𝑠=𝔼_𝑇²+2𝔼^𝑇₀𝑦^𝑛_𝑠𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠𝑑𝑠+2𝔼^𝑇₀𝑦^𝑛_𝑠𝑑𝐴^𝑛_𝑠≤𝔼𝑌_𝑇²+2𝔼^𝑇₀𝑦^𝑛_𝑠𝑔(𝑠,0,0)+𝑢_𝑠𝜑𝑦^𝑛_𝑠+𝑣_𝑠𝜓𝑧^𝑛_𝑠𝑑𝑠+2𝔼^𝑇₀𝑦^𝑛_𝑠𝑑𝐴^𝑛_𝑠. (28) Then, the Hölder inequality and the inequality𝑎𝑏≤𝜖𝑎²+1/𝜖𝑏², for all𝑎,𝑏,𝜖>0, imply that 𝑦^𝑛₀²+𝔼^𝑇₀𝑧^𝑛_𝑠²𝑌𝑑𝑠≤𝔼_𝑇²+1+𝑏_𝜑+𝑏_𝜓+2𝑎_𝜑^𝑇₀𝑢_𝑠𝑑𝑠+2𝑎²_𝜓^𝑇₀𝑣²_𝑠𝑑𝑠𝔼sup_{0≤𝑡≤𝑇}𝑦^𝑛_𝑠²+𝔼^𝑇₀𝑔(𝑠,0,0)𝑑𝑠²+𝑏_𝜑^𝑇₀𝑢_𝑠𝑑𝑠²+𝑏_𝜓^𝑇₀𝑣_𝑠𝑑𝑠²+12𝔼^𝑇₀𝑧^𝑛_𝑠²+1𝑑𝑠4𝐶₁𝔼𝐴^𝑛_𝑇²+4𝐶₁𝔼sup_{0≤𝑡≤𝑇}𝑦^𝑛_𝑠². (29) Thus, we can choose a constant𝐶₂satisfying 𝔼^𝑇₀𝑧^𝑛_𝑠²𝑑𝑠≤𝐶₂+12𝐶₁𝔼𝐴^𝑛_𝑇². (30) Combining the inequalities (27) and (30), we can conclude that 𝔼|𝐴^𝑛_𝑇|²≤2𝐶₁(1+𝐶₂) and 𝔼∫^𝑇₀|𝑧^𝑛_𝑠|²𝑑𝑠≤1+2𝐶₂. The proof is completed.

Then, we give a proposition which plays a key role in the procedure to prove the main theorem.

Proposition 11. lim_𝑛→∞𝔼sup_{0≤𝑡≤𝑇}𝑌_𝑡−𝑦^𝑛_𝑡²=0. (31)

Proof. Since the family of the processes(𝑦^𝑛_𝑡)_{𝑡∈[0,𝑇]}is increasing in𝑛 and dominated by the process(𝑌_𝑡)_{𝑡∈[0,𝑇]}from the above, we can define a process(𝑦_𝑡)_{𝑡∈[0,𝑇]}pointwise by the limit of the processes sequence. Then we have, P-a.s., 𝑦_𝑡∶=lim_𝑛→∞𝑦^𝑛_𝑡,∀𝑡∈[0,𝑇]. (32)
And according to Lemma 2, for any integer𝑛, the following BSDE has a unique solution, denoted by(𝑦^𝑛_𝑡,𝑧^𝑛_𝑡)_{𝑡∈[0,𝑇]}: 𝑦^𝑛_𝑡=𝑌_𝑇+^𝑇_𝑡𝑔𝑠,𝑦^𝑛_𝑠,𝑦^𝑛_𝑠−𝑑𝑠_𝑇𝑡𝑧^𝑛_𝑠𝑑𝐵_𝑠+𝑛^𝑇_𝑡𝑣²_𝑠𝑌_𝑠−𝑦^𝑛_𝑠𝑑𝑠. (33) Let𝜏be a stopping time such that0≤𝜏≤𝑇; then we have 𝑦^𝑛_𝜏=𝔼^ℱ𝜏𝑒^{−𝑛∫𝑇𝜏𝑣2𝑟𝑑𝑟}𝑌_𝑇+𝑛^𝑇_𝜏𝑣²_𝑠𝑒^{−𝑛∫𝑠𝜏𝑣2𝑟𝑑𝑟}𝑌_𝑠+𝑑𝑠_𝑇𝜏𝑒^{−𝑛∫𝑠𝜏𝑣2𝑟𝑑𝑟}𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠𝑑𝑠. (34) For the first two terms within the bracket on the right-hand side of (34), with the property of the vague convergence for the distribution functions, it is easily seen that 𝑒^{−𝑛∫𝑇𝜏𝑣2𝑟𝑑𝑟}𝑌_𝑇+𝑛^𝑇_𝜏𝑣²_𝑠𝑒^{−𝑛∫𝑠𝜏𝑣2𝑟𝑑𝑟}𝑌_𝑠𝑑𝑠⟶𝑌_𝜏,P-a.s., (35) and then, by dominated convergence, it converges in mean square; that is, 𝔼𝑒^{−𝑛∫𝑇𝜏𝑣2𝑟𝑑𝑟}𝑌_𝑇+𝑛^𝑇_𝜏𝑣²_𝑠𝑒^{−𝑛∫𝑠𝜏𝑣2𝑟𝑑𝑟}𝑌_𝑠𝑑𝑠−𝑌_𝜏²⟶0. (36) Now, we come to treat the third term. From the assumption (H2), we can deduce that ^𝑇_𝜏𝑒^{−𝑛∫𝑠𝜏𝑣2𝑟𝑑𝑟}𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠𝑑𝑠≤𝑎_𝜓^𝑇_𝜏𝑒^{−𝑛∫𝑠𝜏𝑣2𝑟𝑑𝑟}𝑣_𝑠𝑧^𝑛_𝑠+𝑑𝑠_𝑇𝜏𝑒^{−𝑛∫𝑠𝜏𝑣2𝑟𝑑𝑟}𝑔(𝑠,0,0)+𝑎_𝜑𝑢(𝑠)𝑦^𝑛_𝑠+𝑏_𝜑𝑢_𝑠+𝑏_𝜓𝑣_𝑠𝑑𝑠. (37) For the integrand of the second integration term on the right hand of (37), it is dominated by 𝑃_𝑡∶=𝑔(𝑡,0,0)+𝑎_𝜑𝑢(𝑡)𝑦¹_𝑡+𝑌_𝑡+𝑏_𝜑𝑢_𝑡+𝑏_𝜓𝑣_𝑡. (38) Combining the assumption (H1), and the fact that(𝑦¹_𝑡)_{𝑡∈[0,𝑇]} and (𝑌_𝑡)_{𝑡∈[0,𝑇]}belong to the space𝒮²(0,𝑇;𝑅), we can obtain that this term converges to zero almost surely with respect to probability𝑃, by dominated convergence theorem, and then 𝔼^𝑇_𝜏𝑒^{−𝑛∫𝑠𝜏𝑣2𝑟𝑑𝑟}𝑔(𝑠,0,0)+𝑎_𝜑𝑢(𝑠)𝑦^𝑛_𝑠+𝑏_𝜑𝑢_𝑠+𝑏_𝜓𝑣_𝑠𝑑𝑠²⟶0. (39) Applying Hölder’s inequality to the first term on the right hand of (37), we can get ^𝑇_𝜏𝑒^{−𝑛∫𝑠𝜏𝑣2𝑟𝑑𝑟}𝑣_𝑠𝑧^𝑛_𝑠1𝑑𝑠≤√2𝑛^𝑇𝜏𝑧^𝑛_𝑠²𝑑𝑠^1/2. (40) Thus, from Proposition 10, it is easy to obtain the following convergence: 𝔼^𝑇_𝜏𝑒^{−𝑛∫𝑠𝜏𝑣2𝑟𝑑𝑟}𝑣_𝑠𝑧^𝑛_𝑠𝑑𝑠²≤12𝑛𝐶⟶0, (41) and then 𝔼^𝑇_𝜏𝑒^{−𝑛∫𝑠𝜏𝑣2𝑟𝑑𝑟}𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠𝑑𝑠²⟶0. (42) Consequently, using Jensen’s inequality and the property of conditional expectation, we have𝔼[(𝑦^𝑛_𝜏−𝑌_𝜏)²]→0.
According to the uniqueness of the solutions for BSDE (17) and the definition (32), we can obtain𝑦^𝑛_𝑡=𝑦^𝑛_𝑡, for all 𝑡∈[0,𝑇], P-a.s., and 𝑦_𝜏=𝑌_𝜏. By section theorem, we have, P-a.s., 𝑦_𝑡=𝑌_𝑡,∀𝑡∈[0,𝑇]. (43)
Therefore, if 𝑇<∞, that𝑌_𝑡−𝑦^𝑛_𝑡uniformly converges to zero in𝑡almost surely with respect to probability𝑃, is the immediate result of Dini’s theorem. Otherwise𝑇=∞, since the increasing sequence of the continuous process(𝑌_𝑡−𝑦^𝑛_𝑡)_{𝑡∈[0,𝑇]}has the same value0at𝑇; then almost surely, for any𝑛and𝜖>0, we can choose a real number𝑀, which may depend only on𝜖and𝜔, such that if𝑡>𝑀, then 𝑌_𝑡−𝑦^𝑛_𝑡≤𝑌_𝑡−𝑦¹_𝑡≤𝜖. (44) On the other hand, by Dini’s theorem,(𝑌_𝑡−𝑦^𝑛_𝑡)_{𝑡∈[0,𝑇]}converges uniformly to zero almost surely on the interval[0,𝑀]. So we can choose a number𝑁depending only on𝜖 and 𝜔such that if𝑛>𝑁, then 𝑌_𝑡−𝑦^𝑛_𝑡≤𝜖,∀𝑡∈[0,𝑀]. (45) Thus,(𝑌_𝑡−𝑦^𝑛_𝑡)_{𝑡∈[0,𝑇]}uniformly converges to zero on the whole interval[0,𝑇]almost surely with respect to probability𝑃. Noticing the fact that|𝑌_𝑡−𝑦^𝑛_𝑡|≤|𝑌_𝑡|+|𝑦¹_𝑡|, we can obtain the desired result by dominated convergence theorem. The proof is completed.

After that, we can get the following proposition about the two sequences of(𝑧^𝑛_𝑡)_{𝑡∈[0,𝑇]} and (𝐴^𝑛_𝑡)_{𝑡∈[0,𝑇]}parameterized by𝑛.

Proposition 12. The processes(𝑧^𝑛_𝑡)_{𝑡∈[0,𝑇]} and (𝐴^𝑛_𝑡)_{𝑡∈[0,𝑇]}, at least their subsequences, are the Cauchy sequences inℋ²(0,𝑇;ℝ^𝑑) and 𝒮²(0,𝑇;ℝ), respectively.

Proof. Applying Itô’s formula to|𝑦^𝑛_𝑡−𝑦^𝑚_𝑡|²on[0,𝑇], we can obtain 𝑦^𝑛₀−𝑦^𝑚₀²+^𝑇₀𝑧^𝑛_𝑠−𝑧^𝑚_𝑠²𝑑𝑠=2^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠−𝑔𝑠,𝑦^𝑚_𝑠,𝑧^𝑚_𝑠𝑑𝑠+2^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑑𝐴^𝑛_𝑠−𝐴^𝑚_𝑠−2^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑧^𝑛_𝑠−𝑧^𝑚_𝑠𝑑𝐵_𝑠≤2^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠−𝑔𝑠,𝑦^𝑚_𝑠,𝑧^𝑚_𝑠𝑑𝑠+2^𝑇₀𝑌_𝑠−𝑦^𝑛_𝑠𝑑𝐴^𝑚_𝑠+2^𝑇₀𝑌_𝑠−𝑦^𝑚_𝑠𝑑𝐴^𝑛_𝑠−2^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑧^𝑛_𝑠−𝑧^𝑚_𝑠𝑑𝐵_𝑠. (46) Due to the fact that the part of Itô integration is uniformly integrable martingale, we have 𝔼^𝑇₀𝑧^𝑛_𝑠−𝑧^𝑚_𝑠²𝑑𝑠≤2𝔼^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠−𝑔𝑠,𝑦^𝑚_𝑠,𝑧^𝑚_𝑠𝑑𝑠+2𝔼^𝑇₀𝑌_𝑠−𝑦^𝑛_𝑠𝑑𝐴^𝑚_𝑠+2𝔼^𝑇₀𝑌_𝑠−𝑦^𝑚_𝑠𝑑𝐴^𝑛_𝑠. (47) As for the last two terms of the above inequality, Propositions 10 and 11 lead to the fact that if𝑚,𝑛→∞, then 𝔼^𝑇₀𝑌_𝑠−𝑦^𝑛_𝑠𝑑𝐴^𝑚_𝑠+𝔼^𝑇₀𝑌_𝑠−𝑦^𝑚_𝑠𝑑𝐴^𝑛_𝑠⟶0. (48) Next, we will show that, as𝑚,𝑛→∞, 𝔼^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠−𝑔𝑠,𝑦^𝑚_𝑠,𝑧^𝑚_𝑠𝑑𝑠⟶0. (49) Because the generator𝑔satisfies the assumption (H2), by Hölder’s inequality, we have 𝔼^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠−𝑔𝑠,𝑦^𝑚_𝑠,𝑧^𝑚_𝑠𝑑𝑠≤𝔼^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑎_𝜑𝑢_𝑠𝑦^𝑛_𝑠−𝑦^𝑚_𝑠+𝑏_𝜑𝑢_𝑠+𝑎_𝜓𝑣_𝑠𝑧^𝑛_𝑠−𝑧^𝑚_𝑠+𝑏_𝜓𝑣_𝑠𝑑𝑠≤𝑎_𝜓𝔼^𝑇₀𝑧^𝑛_𝑠−𝑧^𝑚_𝑠²𝑑𝑠^1/2×𝔼^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠²𝑣²_𝑠𝑑𝑠^1/2+𝑏_𝜓𝔼^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑣_𝑠𝑑𝑠+𝑎_𝜑𝔼sup_{0≤𝑠≤𝑇}𝑦^𝑛_𝑠−𝑦^𝑚_𝑠^2𝑇₀𝑢_𝑠𝑑𝑠+𝑏_𝜑𝔼^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑢_𝑠𝑑𝑠. (50) According to Proposition 10 and the algebraic inequality, we can conclude 𝔼^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠−𝑔𝑠,𝑦^𝑚_𝑠,𝑧^𝑚_𝑠𝑑𝑠≤2𝐶²𝔼^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠²𝑣²_𝑠𝑑𝑠^1/2+𝑏_𝜑𝔼^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑢_𝑠𝑑𝑠+𝑎_𝜑𝔼sup_{0≤𝑠≤𝑇}𝑦^𝑛_𝑠−𝑦^𝑚_𝑠^2𝑇₀𝑢_𝑠𝑑𝑠+𝑏_𝜓𝔼^𝑇₀𝑦^𝑛_𝑠−𝑦^𝑚_𝑠𝑣_𝑠𝑑𝑠. (51) Now set𝐺_𝑠=|𝑦¹_𝑠|+|𝑌_𝑠|,𝐻_𝑠=4𝐺²_𝑠𝑣²_𝑠, and 𝐹_𝑠=2𝐺_𝑠𝑢_𝑠; then, for any𝑚,𝑛≥1, we have 𝑣²_𝑠𝑦^𝑛_𝑠−𝑦^𝑚_𝑠²≤𝐻_s,𝑢_𝑠𝑦^𝑛_𝑠−𝑦^𝑚_𝑠≤𝐹_𝑠[];𝔼,𝑠∈0,𝑇_𝑇0𝐻_𝑠𝑑𝑠≤4𝔼sup_{0≤𝑠≤𝑇}𝐺_𝑠^2𝑇₀𝑣²_𝑠𝔼𝑑𝑠<+∞;_𝑇0𝐹_𝑠𝑑𝑠≤2𝔼sup_{0≤𝑠≤𝑇}𝐺_𝑠^21/2𝑇₀𝑢_𝑠𝑑𝑠<+∞. (52) The first two terms of the right-hand side of (51) converge to zero by the Lebesgue dominated theorem. And Proposition 11 implies that(𝑦^𝑛_𝑡)_{𝑡∈[0,𝑇]}is a Cauchy sequence in𝒮²(0,𝑇;ℝ); then the third term converges to zero. The convergence of the last term can be proved in a similar way to the second one.
Now, coming back to the inequality (47), we can conclude that 𝔼∫^𝑇₀|𝑧^𝑛_𝑠−𝑧^𝑚_𝑠|²𝑑𝑠→0. This meansthat(𝑧^𝑛_𝑡)_{𝑡∈[0,𝑇]}is a Cauchy sequence inℋ²(0,𝑇;ℝ^𝑑), and we denoted its limit by(𝑧_𝑡)_{𝑡∈[0,𝑇]}.
From (17), we know that 𝐴^𝑛_𝑡−𝐴^𝑚_𝑡=𝑦^𝑛₀−𝑦^𝑚₀−^𝑡₀𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠−𝑔𝑠,𝑦^𝑚_𝑠,𝑧^𝑚_𝑠+𝑑𝑠_𝑡0𝑧^𝑛_𝑠−𝑧^𝑚_𝑠𝑑𝐵_𝑠, (53) and then, from the basic algebraic inequality and BDG’s inequality, we can get 𝔼sup_{0≤𝑡≤𝑇}𝐴^𝑛_𝑡−𝐴^𝑚_𝑡²≤3𝔼𝑦^𝑛₀−𝑦^𝑚₀²+3𝔼^𝑇₀𝑧^𝑛_𝑠−𝑧^𝑚_𝑠²𝑑𝑠+3𝔼^𝑇₀𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠−𝑔𝑠,𝑦^𝑚_𝑠,𝑧^𝑚_𝑠𝑑𝑠². (54) In order to show that, when𝑚,𝑛→∞, the limit of the third term of the right-hand side of (54) is zero, we only need to show that if𝑛→∞, then 𝔼^𝑇₀𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠−𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠²⟶0. (55) Because(𝑧^𝑛_𝑡)_{𝑡∈[0,𝑇]}is a Cauchy sequence inℋ²(0,𝑇;ℝ^𝑑), there is at least a subsequence(𝑧^𝑛𝑘_𝑡)_{𝑡∈[0,𝑇]}such that𝑑𝑃×𝑑𝑡-a.e., 𝑧^𝑛𝑘_𝑡→𝑧_𝑡, and 𝑧̆_𝑡∶=sup_𝑘≥1|𝑧^𝑛𝑘_𝑡|∈ℋ²(0,T;ℝ^𝑑). For convenience, we denote the subsequence by(𝑧^𝑛_𝑡)_{𝑡∈[0,𝑇]}itself. According to the assumption (H2), we can deduce that 𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠−𝑔𝑠,𝑦_𝑠,𝑧_𝑠≤𝑎_𝜑𝑢_𝑠𝑦^𝑛_𝑠−𝑦_𝑠+𝑏_𝜑𝑢_𝑠+𝑎_𝜓𝑣_𝑠𝑧^𝑛_𝑠−𝑧_𝑠+𝑏_𝜓𝑣_𝑠. (56) The right-hand side of the above inequality is dominated by 𝑅_𝑠∶=𝑎_𝜑𝑢_𝑠𝐺_𝑠+𝑏_𝜑𝑢_𝑠+𝑎_𝜓𝑣_𝑠𝑧̆_𝑠+𝑧_𝑠+𝑏_𝜓𝑣_𝑠. (57) It is easy to check that𝔼[(∫^𝑇₀𝑅_𝑠𝑑𝑠)²]<∞. Then the convergence of (55) is a direct consequence of the Lebesgue dominated convergence theorem.
From the above argument and Proposition 11, we can assert that(𝐴^𝑛_𝑡)_{𝑡∈[0,𝑇]}is also a Cauchy sequence in𝒮²(0,𝑇;ℝ)with a unique limit(𝐴_𝑡)_{𝑡∈[0,𝑇]}. The proof is completed.

Proof of Theorem 8. From the procedure of the proof of Proposition 12, we know that ^𝑇_𝑡𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠𝑑𝑠⟶^𝑇_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠, (58) uniformly on  [0,T],  in mean square,  that is, 𝔼sup_{0≤𝑡≤𝑇}^𝑇_𝑡𝑔𝑠,𝑦^𝑛_𝑠,𝑧^𝑛_𝑠𝑑𝑠−^𝑇_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠²⟶0. (59) And, by the property of Itô’s integration, BDG’s inequality, and Proposition 12, we also have ^𝑇_𝑡𝑧^𝑛_𝑠𝑑𝐵_𝑠⟶^𝑇_𝑡𝑧_𝑠𝑑𝐵_𝑠, (60) uniformly on [0,T],  in mean square,  that is, 𝔼sup_{0≤𝑡≤𝑇}^𝑇_𝑡𝑧^𝑛_𝑠𝑑𝐵_𝑠−^𝑇_𝑡𝑧_𝑠𝑑𝐵_𝑠²≤𝐶𝔼^𝑇₀𝑧^𝑛_𝑠−𝑧_𝑠²𝑑𝑠⟶0. (61)
Then combining the above convergence and the fact that the sequences(𝑦^𝑛_𝑡)_{𝑡∈[0,𝑇]} and (𝐴^𝑛_𝑡)_{𝑡∈[0,𝑇]}themselves or their subsequences converge to(𝑦_𝑡)_{𝑡∈[0,𝑇]}and(𝐴_𝑡)_{𝑡∈[0,𝑇]}uniformly on[0,𝑇], in mean square, respectively, we can obtain the following equation: 𝑦_𝑡=𝑌_𝑇+∫^𝑇_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠−∫^𝑇_𝑡𝑧_𝑠𝑑𝐵_𝑠+𝐴_𝑇−𝐴_𝑡,𝑡∈[0,𝑇]. (62) Notice that(𝐴^𝑛_𝑡)_{𝑡∈[0,𝑇]}is an increasing process with𝐴^𝑛₀=0; then its limit(𝐴_𝑡)_{𝑡∈[0,𝑇]}will preserve this property. In fact, we have proved the first part of Theorem 8, because, according to (43), the𝑔-supermartingale(𝑌_𝑡)_{𝑡∈[0,𝑇]}coincides with the first component(𝑦_𝑡)_{𝑡∈[0,𝑇]}of the solution for the BSDE (62). And finally, the uniqueness of the decomposition of𝑔-supermartingale follows from the uniqueness of the decomposition of𝑔-supersolution. The proof is completed.

Now, in addition, if we assume that𝑔is independent of𝑦, then we can write the decomposition of Doob-Meyer’s type for𝑔-supermartingale in a more clear sense like the classical martingale theory.

Corollary 13. Let𝑔be independent of𝑦and satisfy the conditions (H1) and (H2). If(𝑋_𝑡)_{𝑡∈[0,𝑇]}is a continuous𝑔-supermartingale in𝒮²(0,𝑇;ℝ), then it has the following decomposition: 𝑋_𝑡=𝑀_𝑡−𝐴_𝑡, (63) where(𝑀_𝑡)_{𝑡∈[0,𝑇]}is a𝑔-martingale and(𝐴_𝑡)_{𝑡∈[0,𝑇]}is an increasing process which belongs to𝒜²(0,𝑇;ℝ).

Proof. By Theorem 8, a𝑔-supermartingale(𝑋_𝑡)_{𝑡∈[0,𝑇]}on[0,𝑇]has the following form. There exists a pair of processes(𝑧_𝑡,𝐴_𝑡)_{𝑡∈[0,𝑇]}such that 𝑋_𝑡=𝑋_𝑇+∫^𝑇_𝑡𝑔𝑠,𝑧_𝑠𝑑𝑠−∫^𝑇_𝑡𝑧_𝑠𝑑𝐵_𝑠+𝐴_𝑇−𝐴_𝑡,𝑡∈[0,𝑇]. (64) We set𝑀_𝑡=𝑋_𝑡+𝐴_𝑡; then 𝑀_𝑡=𝑋_𝑇+𝐴_𝑇+^𝑇_𝑡𝑔𝑠,𝑧_𝑠𝑑𝑠−^𝑇_𝑡𝑧_𝑠𝑑𝐵_𝑠,𝑡∈[0,𝑇]. (65) Obviously, the pair of the processes(𝑀_𝑡,𝑧_𝑡)_{𝑡∈[0,𝑇]}is a solution of the BSDE with the terminal data𝑋_𝑇+𝐴_𝑇and the generator𝑔. Definition 6 implies that𝑀_𝑡is a𝑔-martingale. The proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was supported by the Fundamental Research Funds for the Central Universities (no. 2013DXS03), the Research Innovation Program for College Graduates of Jiangsu Province (no. CXZZ13_0921), and the National Natural Science Foundation of China (no. 11371162).