二次增长条件下的倒向重随机微分方程及其应用

倒向重随机微分方程被广泛应用于随机偏微分方程、随机控制、金融数学以及机器学习等领域。然而，此前关于倒向重随机微分方程的研究，大多基于Lipschitz线性增长的条件，导致其在更一般条件下的应用显得力不能及。本项目旨在研究二次增长条件下的倒向重随机微分方程，并将其应用于随机控制和随机偏微分方程领域中。具体围绕三个问题展开研究：（1）解的存在唯一性。本项目拟引入Yamada-Watanabe定理的新思想突破瓶颈。在终值有界的条件下，通过比较定理证明强解的唯一性，再结合弱解的存在性进一步证明强解的存在性。（2）在随机最优控制问题中的应用。在二次增长的条件下，建立倒向重随机控制系统的随机最大值原理和随机线性二次最优控制问题。（3）与随机偏微分方程之间的关系。在二次增长的条件下，研究对应随机偏微分方程的经典解和粘性解的存在唯一性，建立相应的非线性随机Feynman-Kac公式。

Backward doubly stochastic differential equations (BDSDEs, for short) have been widely used in many fields, such as stochastic control, stochastic partial differential equations, financial mathematics, and machine learning. However, most works about BDSDEs are based on Lipschitz condition up to now, which makes it is hard to use BDSDEs in applications under general conditions. In this project, we study backward doubly stochastic differential equations with quadratic growth and their applications to stochastic control and stochastic partial differential equations. In detail, we study three main problems: First, the existence and uniqueness of the solution. We will use the Yamada-Watanabe theorem to make a breakthrough. When the terminal value is bounded, we use the comparison theorem to prove the uniqueness of the solutions. Moreover, combining the existence of weak solutions, the Yamada-Watanabe theorem implies the existence of strong solutions. Second, we study the application of BDSDEs with quadratic growth to stochastic optimal control, including the stochastic maximum principle and stochastic linear-quadratic optimal control problem. Third, the relation between BDSDEs with quadratic growth and stochastic partial differential equations will be investigated, the classical solution and viscosity solution of the stochastic partial differential equations, and the related nonlinear Feynman-Kac formula will be established.