勷勤数学•专家报告

题      目：Optimal Controls for Forward-Backward Stochastic Differential Equations: From Pre-Committed Solutions to Equilibrium Solutions

报  告  人： 王寒霄 教授  (邀请人：杨舟)

                                          深圳大学

时      间： 3月21日  10：00-11:00

地     点：数科院西楼二楼会议室

报告人简介:

        王寒霄，深圳大学数学科学学院教授，博士生导师。主要从事随机控制理论的研究，关注时间不一致问题、随机 Volterra 积分方程及相关的路径依赖偏微分方程、随机线性二次问题等，结果发表在J. Math. Pures Appl.、SIAM J. Control Optim（3篇）.、Finance Stoch.、Ann. Inst. Henri Poincare Probab. Stat.、J. Differential Equations等期刊。现主持国家自然科学基金青年项目B类等科研项目，独立获得2021年Stochastics and Dynamics最佳论文奖。

摘      要：

       This talk is concerned with an optimal control problem for a forward-backward stochastic differential equation (FBSDE, for short) with a recursive cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). It is found that such an optimal control problem is time-inconsistent in general, even if the cost functional is reduced to a classical Bolza type one as in Peng (AMO 1993), Lim-Zhou (SICON 2001), and Yong (SICON 2010). Therefore, instead of finding a global optimal control (which is time-inconsistent), we will look for a time-consistent and locally optimal equilibrium strategy, which can be constructed via the solution of an associated equilibrium Hamilton-Jacobi-Bellman (HJB, for short) equation. A verification theorem for the local optimality of the equilibrium strategy is proved by means of the generalized Feynman-Kac formula for BSVIEs and some stability estimates of the representation parabolic partial differential equations (PDEs, for short). Under certain conditions, it is proved that the equilibrium HJB equation, which is a nonlocal PDE, admits a unique classical solution. As special cases, the linear-quadratic problem is considered, in which the solvability of a non-local and non-symmetric matrix-valued Riccati equation is established. As applications, a social planner problem with heterogeneous Epstein-Zin utilities is briefly mentioned, by which we show that the Epstein-Zin utility is much more effective than the power utility in this model. 

Joint work with Qi Lü, Bowen Ma, Jiongmin Yong, and Chao Zhou.

          欢迎老师、同学们参加、交流！