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

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


Introduction
In 1990, Pardoux-Peng [1] proposed the following nonlinear backward stochastic differential equation (BSDE) driven by a Brownian motion: (1) where the positive real number , the random variable , and the function  are called the time horizon, the terminal data, and the generator, respectively, and the pair of adapted processes (  ,   ) ∈[0,] 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 (  ) ∈[0,] with  0 = 0: (2) If (  ) ∈[0,] ≡ 0, the first component (  ) ∈[0,] of solution of ( 2) is called the -solution of (1); otherwise, it is called the -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 of -supersolutions is also a -supersolution.And applying this result, he proved that a càdlàg -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 the -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 smallest 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 on , whose Lipschitzian coefficient is a function depending on , and they obtained the convergence theorem of the nonlinear -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 coefficient , does the -martingale still have a nonlinear

Preliminaries
Let  be a finite or infinite nonnegative extended real number, and let (  ) ≥0 be a standard -dimensional Brownian motion defined on a complete probability space (Ω, F, ) endowed with a filtration (F  ) ≥0 generated by this Brownian motion: where N is the set of all -null subsets.
For simplicity of presentation, we use || to denote the Euclidean norm of  in R or R  , and let  2 (Ω, F  , ) be the space of all the F  measurable square integrable real valued random variables, and define the adapted process spaces as follows: Clearly, all the above spaces of stochastic processes are completed Banach spaces.
Furthermore, we denote the set of linear increasing functions (⋅) : R +  → R + with (0) = 0 by X.Here the linear increasing means that, for any element  ∈ X, there exists a pair of positive real numbers (, ) depending on  such that, for all  ∈ R + , () ≤  + .
The generator (, , , ) : [0, ] × Ω × R × R   → R is a random function which is a progressively measurable stochastic process for any (, ).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 on : , where (⋅) and V(⋅) are two positive functions mapping from R + to R + , such that ∫  0 [() + V 2 ()]  < ∞; the functions  and  belong toX and (⋅) is a concave function, with ∫ 0 + ()  = +∞.
And in addition, we assume that ∫  0 V()  < ∞, if  cannot be dominated by a linear function; that is, we cannot find a real number , such that () ≤ .
[9].Then we can prove that, for all  ∈ [0, ], ŷ+  ≤ 0, P-a.s.Therefore, for any  ∈ [0, ], we have Observing that   and   are càdlàg processes, we can conclude that, P-a.s., Remark 4. If we replace the deterministic terminal time  by a F  -stopping time  ≤ , then, by Lemma 2, existence and uniqueness theorem and the above comparison theorem still hold true.
Next, we introduce the conceptions of -solution, supersolution, -martingale, and -supermartingale closely following Peng's definitions in [3].From Proposition 3, we know that a -supersolution is a -supermartingale.Conversely, a meaningful and interesting question follows immediately.Is a -supermartingale a -supersolution?If so, does the -supermartingale, or supersolution, has a unique representation of the form (15)?

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 continuous -supermartingale is a -supersolution; that is, it has a unique decomposition in the sense of Definition 7.
In order to prove this theorem, we consider the family of penalization BSDEs parameterized by  = 1, 2, 3, . .., and set We first claim the next proposition.
Proposition 9.For each  = 1, 2, . .., one has, P-a.s., 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 exist  > 0 and a positive integer  such that the measure of {(, ) |    −   −  ≥ 0} ⊂ Ω × [0, ] is nonzero; then we can define the following stopping times: It is observed, from the above definition and the continuous of (  ) ∈[0,] , that  ≤  ≤  and ( > ) > 0. And furthermore, we have, P-a.s., Thus, we can choose a constant  2 satisfying Combining the inequalities ( 27) and (30), we can conclude that Then, we give a proposition which plays a key role in the procedure to prove the main theorem.
Proof.Since the family of the processes (   ) ∈[0,] is increasing in  and dominated by the process (  ) ∈[0,] from the above, we can define a process (  ) ∈[0,] pointwise by the limit of the processes sequence.Then we have, P-a.s., And according to Lemma 2, for any integer , the following BSDE has a unique solution, denoted by ( ỹ  , z  ) ∈[0,] : Let  be a stopping time such that 0 ≤  ≤ ; then we have 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 →   , P-a.s., (35) and then, by dominated convergence, it converges in mean square; that is, Now, we come to treat the third term.From the assumption (H2), we can deduce that For the integrand of the second integration term on the right hand of (37), it is dominated by Combining the assumption (H1), and the fact that ( 1  ) ∈[0,] and (  ) ∈[0,] belong to the space S 2 (0, ; ), we can obtain that this term converges to zero almost surely with respect to probability , by dominated convergence theorem, and then Applying Hölder's inequality to the first term on the right hand of (37), we can get Thus, from Proposition 10, it is easy to obtain the following convergence: and then Consequently, using Jensen's inequality and the property of conditional expectation, we have E[( ỹ  −   ) 2 ] → 0. According to the uniqueness of the solutions for BSDE (17) and the definition (32), we can obtain    = ỹ  , for all  ∈ [0, ],P-a.s., and   =   .By section theorem, we have, P-a.s., Therefore, if  < ∞, that   −    uniformly converges to zero in  almost surely with respect to probability , is the immediate result of Dini's theorem.Otherwise  = ∞, since the increasing sequence of the continuous process (  −    ) ∈[0,] has the same value 0 at ; then almost surely, for any  and  > 0, we can choose a real number , which may depend only on  and , such that if  > , then The first two terms of the right-hand side of (51) converge to zero by the Lebesgue dominated theorem.And Proposition 11 implies that (   ) ∈[0,] is a Cauchy sequence in S 2 (0, ; 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.] . (54) In order to show that, when ,  → ∞, the limit of the third term of the right-hand side of ( 54 ,   ,   )  − ∫       +   −   ,  ∈ [0, ] .