The relationship between maximum principle and dynamic programming principle for stochastic recursive control problem with random coefficients

¹ School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai 200092, PR China
² Department of Mathematical Sciences, Huzhou University, Zhejiang 313000, PR China
³ School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China
⁴ Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai 200433, PR China

^* Corresponding author: qzh@fudan.edu.cn

Received: 16 July 2024
Accepted: 21 October 2024

Abstract

This paper aims to explore the relationship between maximum principle and dynamic programming principle for stochastic recursive control problem with random coefficients. Under certain regular conditions for the coefficients, the relationship between the Hamiltonian system with random coefficients and stochastic Hamilton–Jacobi–Bellman equation is obtained. It is very different from the deterministic coefficients case since stochastic Hamilton–Jacobi–Bellman equation is a backward stochastic partial differential equation with solution being a pair of random fields rather than a deterministic function. A linear quadratic recursive optimization problem is given as an explicitly illustrated example based on this kind of relationship.