Relationship between MP and DPP for Risk-Sensitive Stochastic Optimal Control Problems: Viscosity Solution Framework

TL;DR

This paper explores the connection between maximum principle and dynamic programming in risk-sensitive stochastic control, using viscosity solutions to handle non-convex control domains and recursive systems.

Contribution

It establishes theoretical relations between adjoint processes, Hamiltonian functions, and value functions in a viscosity solution framework for complex control problems.

Findings

Proves equivalence between maximum principle and dynamic programming for risk-sensitive control.

Derives relations among adjoint processes, Hamiltonian, and value function.

Provides examples illustrating the theoretical results.

Abstract

In this paper, we study the relationship between general maximum principle and dynamic programming principle for risk-sensitive stochastic optimal control problems, where the control domain is not necessarily convex. The original problem is equivalent to a stochastic recursive optimal control problem of a forward-backward system with quadratic generators. Relations among the adjoint processes, the generalized Hamiltonian function and the value function are proved under the framework of viscosity solutions. Some examples are given to illustrate the theoretical results.