On weak solutions of forward–backward SDEs
On weak solutions of forward–backward SDEs
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前向-后向 SDE 的弱解
DOI：
10.1007/s00440-010-0305-8
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发表时间：
2011
影响因子：
2
通讯作者：
Jianfeng Zhang
中科院分区：
文献类型：
--
作者：
Jin Ma;Jianfeng Zhang
本文继续研究正倒向随机微分方程的弱解的概念及相应的正倒向鞅问题。这项工作的主要目的是消除在我们以前的工作（马等）的唯一性证明鞅被积的约束。Ann Probab 36（6）：2092-2125，2008）。我们考虑了一类非退化的FBSDES，其中所有的系数被假定为本质上只有界且一致连续，并且在所有平方可积适应解的空间（FBSDES文献中的标准解空间）中证明了唯一性.引入了半强解的概念，澄清了文献中弱解定义之间的关系，并在唯一性证明中起到了一定的作用。作为副产品，我们还建立了一些先验估计的解的二阶导数的解耦拟线性偏微分方程。
In this paper we continue exploring the notion of weak solution of forward–backward stochastic differential equations (FBSDEs) and associated forward–backward martingale problems (FBMPs). The main purpose of this work is to remove the constraints on the martingale integrands in the uniqueness proofs in our previous work (Ma et al. in Ann Probab 36(6):2092–2125, 2008). We consider a general class of non-degenerate FBSDEs in which all the coefficients are assumed to be essentially only bounded and uniformly continuous, and the uniqueness is proved in the space of all the square integrable adapted solutions, the standard solution space in the FBSDE literature. A new notion of semi-strong solution is introduced to clarify the relations among different definitions of weak solution in the literature, and it is in fact instrumental in our uniqueness proof. As a by-product, we also establish some a priori estimates of the second derivatives of the solution to the decoupling quasilinear PDE.