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

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.