Research Article | Open Access
Nonlinear Decomposition of Doob-Meyer's Type for Continuous -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.
In 1990, Pardoux-Peng  proposed the following nonlinear backward stochastic differential equation (BSDE) driven by a Brownian motion: where the positive real number, the random variable, and the functionare called the time horizon, the terminal data, and the generator, respectively, and the pair of adapted processesto be known is called the solution of the BSDE (1). In this paper, we study a more generalized BSDE with a given increasing processwith: If, the first componentof solution of (2) is called the-solution of (1); otherwise, it is called the-supersolution. Subsequently, Peng  introduced the nonlinear expectation and nonlinear martingale theories via BSDEs. In , 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  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 , 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  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  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.  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 decomposition of Doob-Meyer’s type? Our answer is yes. We prove that if a-supermartingale has a continuous trajectory on finite or infinite time interval, then it is a-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 coefficient. 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 generator. Then, Section 3 devotes to the main result a new version of nonlinear Doob-Meyer’s decomposition theorem for the continuous-supermartingale with the generalized uniformly continuous coefficient.
Letbe a finite or infinite nonnegative extended real number, and letbe a standard-dimensional Brownian motion defined on a complete probability spaceendowed with a filtrationgenerated by this Brownian motion: where is the set of all-null subsets.
For simplicity of presentation, we useto denote the Euclidean norm ofin or , and letbe the space of all themeasurable square integrable real valued random variables, and define the adapted process spaces as follows: is a -valued process such that ; is a càdlàg - valued process such that ; is an increasing process in with .Clearly, all the above spaces of stochastic processes are completed Banach spaces.
Furthermore, we denote the set of linear increasing functionswithby . Here the linear increasing means that, for any element, there exists a pair of positive real numbersdepending onsuch that, for all , .
The generator 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:(H1);(H2), whereandare two positive functions mapping fromto, such that; the functionsandbelong to and is a concave function, with. And in addition, we assume that, ifcannot be dominated by a linear function; that is, we cannot find a real number, such that.
Remark 1. In (H2), is a concave function which means that, forand,. And the equalitymeans that the value of the integrationwill be infinite on any intervalwith. For simplicity, we also useandto denoteand, respectively, in the remaining of this paper.
Now, we consider the following problem. Suppose that the time horizon, generator, terminal data, and the increasing càdlàg process are given in advance; let us find a pair of processessatisfying
If , 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. . Otherwise we can setand treat the following BSDE as It is a classical BSDE with the terminal dataand the generator. Since the assumptions (H1) and (H2) hold for generator, it is easy to verify thatstill satisfies the two conditions. So we have the following existence and uniqueness theorem.
Lemma 2 (existence and uniqueness). One assumes that the generatorof the BSDE (4) satisfies the conditions (H1) and (H2). Then, for any random variable, and a process, there exists a unique pair of processes, which is a solution of the BSDE (4), such thatis continuous and
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. Letbe the solution of another BSDE: where,, and are given such that(1); (2), .;(3) is a càdlàg increasing process;(4).Then we have, P-a.s.,
Proof. We sketch the proof as follows. Set and ; applying Itô-Meyer’s formula toleads to
Sinceis an increasing process, we see that
Recalling that., and the assumption (H2), we can get Thus, it follows that Now we are in the same position with Theorem 2 in Fan et al. . Then we can prove that, for all ,, P-a.s. Therefore, for any, we have Observing thatandare càdlàg processes, we can conclude that, P-a.s.,
Remark 4. If we replace the deterministic terminal timeby a-stopping time, then, by Lemma 2, existence and uniqueness theorem and the above comparison theorem still hold true.
For a given stopping time, we now consider the following BSDE: where and is a given càdlàg increasing process with.
Next, we introduce the conceptions of-solution,-supersolution,-martingale, and -supermartingale closely following Peng’s definitions in .
Definition 5. If a processcan be written in the form of the BSDE (15) with the generator, then one call it a g-supersolution on. Particularly, ifon, then one calla-solution on.
Definition 6. A-progressively measurable real-valued processis called a-supermartingale (resp.-martingale), if for each stopping time, , and the-solutionwith terminal conditionsatisfiesresp. for all stopping time. Indeed, a-martingale onis a-solution on.
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)?
According to Proposition 1.6 in , we can assert that, given a-supersolutionon, there is a unique pair of processes onsuch that the triplesatisfies the BSDE (15). Now, we can propose the next conception as follows.
Definition 7. Provided that the processis a-supersolution and the triple of processessatisfies the BSDE (15), one callthe unique decomposition of.
3. Nonlinear Doob-Meyer’s Decomposition for -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.
Theorem 8. One assumes thatsatisfies the conditionsand. Letbe a continuous-supermartingale onin. Thenis a-supersolution on that is, there is a unique pair of processesin, such thatcoincides with the first componentof the solution for the following BSDE:
In order to prove this theorem, we consider the family of penalization BSDEs parameterized by, and set We first claim the next proposition.
Proposition 9. For each , one has, P-a.s.,
Proof. Using an argument similar to that in Lemma 3.4 in , one can carry out the proof by contradiction. We sketch it as follows.
Supposing that it is not the case, then there existand a positive integersuch that the measure ofis nonzero; then we can define the following stopping times: It is observed, from the above definition and the continuous of, that and . And furthermore, we have, P-a.s., Now let(resp.) be the-solution onwith terminal condition(resp.). By Proposition 3, (21)-(ii) implies that. On the other hand sinceis a-supermartingale, thus we can get This is a contradiction to (21)-(i). Then by Fubini’s theorem, we have, P-a.s., And the conclusion follows from the continuity of. The proof is completed.
Now, we can get the following result; the boundedness of the triple of the processescan be defined by the penalization BSDEs.
Proposition 10. There exists a positive real number C such that for any positive integer
Proof. From BSDE (17), we have
where the real numbers, and ,depend on the functions and , respectively. From Proposition 9, we see thatis dominated by. Thus there exists a constantindependent of, such that
Now, noticing the boundedness ofin the above sense, from the basic algebraic inequality, Jensen’s inequality and Hölder’s inequality, we can get that there exists another constantsuch that
On the other hand, in the light of (H2), applying Itô’s formula toonwill lead to Then, the Hölder inequality and the inequality, for all,,, imply that Thus, we can choose a constantsatisfying Combining the inequalities (27) and (30), we can conclude that and . The proof is completed.
Then, we give a proposition which plays a key role in the procedure to prove the main theorem.
Proof. Since the family of the processesis increasing in and dominated by the processfrom the above, we can define a processpointwise 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: Letbe a stopping time such that; 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 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 and belong to the space, 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.
According to the uniqueness of the solutions for BSDE (17) and the definition (32), we can obtain, for all , -a.s., and . By section theorem, we have, P-a.s.,
Therefore, if , thatuniformly converges to zero inalmost surely with respect to probability, is the immediate result of Dini’s theorem. Otherwise, since the increasing sequence of the continuous processhas the same valueat; then almost surely, for anyand, we can choose a real number, which may depend only onand, such that if, then On the other hand, by Dini’s theorem,converges uniformly to zero almost surely on the interval. So we can choose a numberdepending only on and such that if, then Thus,uniformly converges to zero on the whole intervalalmost surely with respect to probability. Noticing the fact that, 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 and parameterized by.
Proposition 12. The processes and , at least their subsequences, are the Cauchy sequences in and , respectively.
Proof. Applying Itô’s formula toon, we can obtain
Due to the fact that the part of Itô integration is uniformly integrable martingale, we have
As for the last two terms of the above inequality, Propositions 10 and 11 lead to the fact that if, then
Next, we will show that, as,
Because the generatorsatisfies the assumption (H2), by Hölder’s inequality, we have
According to Proposition 10 and the algebraic inequality, we can conclude
Now set,, and ; then, for any,, we have
The first two terms of the right-hand side of (51) converge to zero by the Lebesgue dominated theorem. And Proposition 11 implies thatis a Cauchy sequence in; 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 . This meansis a Cauchy sequence in, and we denoted its limit by.
From (17), we know that and then, from the basic algebraic inequality and BDG’s inequality, we can get In order to show that, when, the limit of the third term of the right-hand side of (54) is zero, we only need to show that if, then Becauseis a Cauchy sequence in, there is at least a subsequencesuch that-a.e., , and . For convenience, we denote the subsequence byitself. According to the assumption (H2), we can deduce that The right-hand side of the above inequality is dominated by It is easy to check that. 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 thatis also a Cauchy sequence inwith a unique limit. The proof is completed.
Proof of Theorem 8. From the procedure of the proof of Proposition 12, we know that
uniformly on [0,T], in mean square, that is,
And, by the property of Itô’s integration, BDG’s inequality, and Proposition 12, we also have
uniformly on [0,T], in mean square, that is,
Then combining the above convergence and the fact that the sequences and themselves or their subsequences converge toanduniformly on, in mean square, respectively, we can obtain the following equation: Notice thatis an increasing process with; then its limitwill preserve this property. In fact, we have proved the first part of Theorem 8, because, according to (43), the-supermartingalecoincides with the first componentof the solution for the BSDE (62). And finally, the uniqueness of the decomposition of-supermartingale follows from the uniqueness of the decomposition of-supersolution. The proof is completed.
Now, in addition, if we assume thatis independent of, then we can write the decomposition of Doob-Meyer’s type for-supermartingale in a more clear sense like the classical martingale theory.
Corollary 13. Letbe independent ofand satisfy the conditions (H1) and (H2). Ifis a continuous-supermartingale in, then it has the following decomposition: whereis a-martingale andis an increasing process which belongs to.
Proof. By Theorem 8, a-supermartingaleonhas the following form. There exists a pair of processessuch that We set; then Obviously, the pair of the processesis a solution of the BSDE with the terminal dataand the generator. Definition 6 implies thatis a-martingale. The proof is completed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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).
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