Optimal Controls for Forward-Backward Stochastic Differential Equations: Time- Inconsistency and Time-Consistent Solutions

报告人：王寒霄 助理教授（深圳大学）

时 间：2024年4月8日 15：00-16：00

地 点：博学楼506会议室、腾讯会议（会议号：448380713）

主办单位：贵州大学数学与统计学院

Abstract

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 and applications, the linear-quadratic problems, a mean-variance model, a social planner problem with heterogeneous Epstein-Zin utilities, and a Stackelberg game are briefly investigated，which show that the introduction of controlled backward state processes is necessary in some cases. We remark that our framework can cover not only the optimal control problems for FBSDEs studied in Peng (AMO 1993), Lim-Zhou (SICON 2001), Yong (SICON 2010), and so on, but also the problems of the general discounting and some nonlinear appearance of conditional expectations for the terminal state, studied in Yong (MCRF 2012, ICM 2014) and Bjork-Khapko-Murgoci (FS 2017). Joint work with Prof. Jiongmin Yong (UCF) and Prof. Chao Zhou (NUS).

个人简介

王寒霄，2014年本科毕业于吉林大学，2020年在雍炯敏教授指导下于复旦大学获得博士学位。2017年10月至2019年5月在美国中佛罗里达大学联合培养。2020年9月至2022年2月在新加坡国立大学数学系任Research Fellow。现就职深圳大学数学科学学院，任助理教授、硕士生导师。主要从事随机控制理论及应用的研究。已在Ann. Inst. Henri Poincare Probab. Stat.、ESAIM COCV（2篇）、Finance Stoch、J. Differential Equations、SIAM J. Control Optim.（2篇）等期刊发表论文13篇。主持国家自然科学基金青年项目、广东省基金面上项目各一项。入选深圳市鹏城孔雀计划，任美国数学会评论员、广东省运筹学会理事、SIAM J. Control Optim., Finance Stoch等期刊审稿人。