The Relationship between Maximum Principle and Dynamic Programming Principle for Stochastic Recursive Control Problem with Random Coefficients

TL;DR

This paper investigates the connection between maximum principle and dynamic programming in stochastic recursive control problems with random coefficients, highlighting differences from deterministic cases and illustrating with a linear quadratic example.

Contribution

It establishes the relationship between Hamilton systems and stochastic Hamilton-Jacobi-Bellman equations for problems with random coefficients, extending classical results to stochastic settings.

Findings

Derived the relationship between Hamilton system and stochastic HJB equation with random coefficients.

Showed that stochastic HJB is a backward stochastic PDE with random solution fields.

Provided an explicit example using a linear quadratic recursive utility optimization problem.

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 Hamilton 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 utility optimization problem is given as an explicitly illustrated example based on this kind of relationship.