几类倒向双重随机微分方程及其应用研究

This project aims to study some classes of backward doubly stochastic differential equations and their applications. Our research work mainly includes the following detailed contents and aims. Firstly, we will give the probabilistic interpretation in the sense of stochastic viscosity solution and weak solution in the Sobolev space for the obstacle problem for a class of stochastic partial differential (-integral) equations by means of reflected backward doubly stochastic differential equations (driven by Lévy process). Moreover, we will propose a general discrete approximation scheme for multi-dimensional reflected backward doubly stochastic differential equations driven by two Brownian motions. Then, an application to obstacle problem with Neumann condition for stochastic partial differential equations will be given. Secondly, we will introduce a new kind of reflected backward doubly stochastic differential equations, where the drift is the nonlinear function of the barrier process. We will establish the existence and uniqueness of the solution for this kind of equations by means of the stochastic variant Skorohod problem based on the stochastic representation theorem and the contraction mapping theorem. The application to optimal stopping problems will also be given. Thirdly, a class of generalized backward doubly stochastic differential equations whose coefficient contains the subdifferential operators of two convex functions, which are also called as generalized backward doubly stochastic variational inequalities, will be considered. By means of a penalization argument based on Yosida approximation, we will establish the existence and uniqueness of the solution. As an application, this result will be used to derive the result of stochastic viscosity solution for a class of multivalued stochastic Dirichlet-Neumann problems. Fourthly, we will deal with a class of backward doubly stochastic differential equations with time delayed coefficients. We will prove the existence and uniqueness of a solution for a sufficient small Lipschitz constant of the coefficients. Moreover, we will give the probabilistic interpretation of stochastic viscosity solution for a class of stochastic partial differential equations with delay. Also, we will establish the stochastic maximum principle for the stochastic optimal control related with the backward doubly stochastic differential equations with time delayed coefficients.

本项目旨在研究几类倒向双重随机微分方程及相关应用问题。主要包括：通过(由Lévy过程驱动的)反射型倒向双重随机微分方程，给出一类具有障碍问题随机偏微分(-积分)方程随机粘性解以及Sobolev空间中一类随机偏微分(-积分)方程弱解的概率表示。构造由两个布朗运动驱动的多维反射型倒向双重随机微分方程的数值解并由此给出一类具有障碍问题随机偏微分方程的数值解；通过随机Skorohod方程研究一般形式的非线性反射型倒向双重随机微分方程，给出其在随机最优停时等问题中的应用；通过惩罚函数法讨论含有两个凸函数次微分算子的广义反射型倒向双重随机微分方程，给出具有Dirichlet-Neumann边界条件的二阶多值随机微分包含随机粘性解的概率表示；通过含有时滞算子倒向双重随机微分方程给出一类具有时滞的随机偏微分方程解的概率表示，建立含有时滞算子倒向双重随机微分方程相应的随机最优控制问题的随机最大值原理。

由于应用的广泛性，倒向随机微分方程备受关注。本课题研究几类倒向双重随机微分方程及相关问题：研究了由Lévy 过程驱动的反射型倒向双重随机微分方程，通过此类方程给出一类具有障碍问题随机偏微分-积分方程随机粘性解的概率表示；探讨了非线性反射型倒向双重随机微分方程。通过随机Skorohod 方程研究一般形式的非线性反射型倒向双重随机微分方程解的存在唯一性，探讨其在随机最优停时等问题中的应用；通过惩罚函数法给出了一类含有两个凸函数次微分算子的广义反射型倒向双重随机微分方程解的存在唯一性，由此给出具有 Dirichlet-Neumann 边界条件的二阶多值随机微分包含随机粘性解的存在唯一性；研究了由马氏链驱动的平均场倒向随机微分方程；通过惩罚函数法等给出了由G-布朗运动驱动的多值倒向随机微分方程方程解的存在唯一性，给出其在一类全非线性偏微分方程解的概率表示方面的应用；通过变分法以及Hamilton 函数等工具建立了一类正-倒向随机微分方程部分可观测的最优控制问题随机最优控制问题的随机最大值原理；给出了多值平均场倒向随机微分方程及其应用；对由G-布朗运动驱动的随机微分方程稳定性进行了系统研究，得到了一些随机系统稳定性的条件；深入探讨了几类随机微分方程的性质及在金融中的应用问题。. 所得研究成果丰富和发展了倒向随机发展方程理论，扩展了其相关应用领域。对近年来发展起来的G-布朗运动驱动的随机微分方程相关控制问题和稳定性进行了系统研究，得到的成果有应用前景。