在SPDE的研究中，倒向随机微分方程的理论和方法在一些问题中起着独到的作用，本项目致力于以倒向随机微分方程为工具，研究几类非线性的SPDE、倒向随机偏微分方程。具体研究内容包括1.证明随机Hamilton-Jacobi-Bellman方程的随机周期解的存在性，并利用对应的控制问题的最优解构造出随机周期解；2.建立随机Hamilton-Jacobi-Bellman方程的非线性Feynman-Kac公式，并应用到随机递归线性二次最优控制问题中；3.建立质量守恒的随机Allen-Cahn方程与对应的倒向重随机微分方程的联系，构造质量守恒的随机Allen-Cahn方程的平稳解；4.证明两类非Lipschitz系数的可退化倒向随机偏微分方程的正则性，并应用到随机微分效用函数的最优消费投资问题中。这些研究内容涉及SPDE的正则性、随机周期解、平稳解等理论，并应用到随机动力系统、随机控制等领域。

In the study of SPDE, the theories and methods of backward stochastic differential equations play a special role in some issues. This project aims to study some types of nonlinear SPDEs, backward stochastic partial differential equations by the tool of backward stochastic differential equations. The specific research contents include: 1. prove the existence of random periodic solution of stochastic Hamilton-Jacobi-Bellman equation, and construct the random periodic solution by the optimal solution of corresponding control problem; 2. establish the nonlinear Feynman-Kac formula for stochastic Hamilton-Jacobi-Bellman equation, and apply it to the stochastic recursive linear quadratic optimal control problem; 3. establish the relationship between mass-conserving stochastic Allen-Cahn equation and corresponding backward doubly stochastic differential equation, and construct the stationary solution of stochastic Allen-Cahn equation; 4. prove the regularities for two types of degenerate backward stochastic partial differential equations with non-Lipschitz coefficients, and apply them to the optimal consumption-investment problems with stochastic differential utilities. The above studies involve the theories like regularities, random periodic solutions and stationary solutions of SPDEs, and have applications in the research fields like random dynamical system, stochastic control.