本项目是最优控制、随机分析和偏微分方程的交叉课题，研究带Poisson跳跃的随机泛函微分方程最优控制及相关的路径依赖积分-偏微分方程。主要是：研究具有有限时区和无限时区的带Poisson跳跃的随机泛函微分方程线性二次最优控制问题，建立相应Riccati方程的可解性条件并给出最优控制的显示表达；研究带Poisson跳跃的随机泛函微分方程最优控制问题的一般动态规划原理，证明值函数满足路径依赖积分-偏微分方程（即路径依赖HJB方程），建立随机验证定理；研究上述路径依赖HJB方程粘性解的存在唯一性，并将理论结果用于研究具有非光滑值函数的带Poisson跳跃的随机泛函微分方程最优控制问题,得到最优反馈控制律。本项目的研究结果不但能推动随机最优控制理论本身的发展，也会促进路径依赖偏微分方程理论的进步。

The project is an intersect project of optimal control, stochastic analysis and partial differential equations, to study the optimal control problem of stochastic functional differential equation with Poisson jump and related path-dependent integro-partial differential equation. The project will mainly study the following problems: The linear-quadratic optimal control problem for stochastic functional differential equation with Poisson jump in finite-time and infinite-time horizon, to establish the solvability conditions of Riccati equation and then characterize the optimal control; The general dynamic programming principle of the optimal control problem for stochastic functional differential equation with Poisson jump, to prove that the value functional satisfies the path-dependent integro-partial differential equations (i.e., the path-dependent HJB equation) and establish the stochastic verification theorem; To study the existence and uniqueness of viscosity solution for the path-dependent HJB equation, and to study the optimal control problem of stochastic functional differential equation with Poisson jump and non-regular value functional, to obtain the law of optimal feedback control. The results of this project will not only improve the stochastic optimal control theory, but also accelerate the development of Path-dependent PDE theory.