Weak Dynamic Programming for Generalized State Constraints

TL;DR

This paper develops a weak dynamic programming approach for stochastic control problems with expectation and state constraints, deriving viscosity solutions to the associated Hamilton-Jacobi-Bellman equations.

Contribution

It introduces a weak formulation that relaxes measurable selection restrictions and applies to both expectation and state constraints, including closed state constraints.

Findings

Provides a dynamic programming principle for expectation constraints

Derives viscosity solutions for HJB equations with state constraints

Establishes a comparison theorem for closed state constraints

Abstract

We provide a dynamic programming principle for stochastic optimal control problems with expectation constraints. A weak formulation, using test functions and a probabilistic relaxation of the constraint, avoids restrictions related to a measurable selection but still implies the Hamilton-Jacobi-Bellman equation in the viscosity sense. We treat open state constraints as a special case of expectation constraints and prove a comparison theorem to obtain the equation for closed state constraints.