Stochastic Optimal Control with Control-Dependent Diffusion and State Constraints: A Degenerate Elliptic Approach

TL;DR

This paper develops a unified mathematical framework for stochastic optimal control problems where the control affects both the drift and degenerate diffusion, with state constraints leading to a complex boundary value problem.

Contribution

It provides the first unified analysis of control-dependent diffusion and state constraints, establishing the uniqueness of viscosity solutions for the associated degenerate elliptic HJB equation.

Findings

Proves the optimal value function is the unique viscosity solution.

Handles control-dependent, possibly degenerate diffusion matrices.

Includes an example demonstrating the framework's applicability.

Abstract

We study a stochastic optimal control problem with the state constrained to a smooth, compact domain. The control influences both the drift and a possibly degenerate, control-dependent dispersion matrix, leading to a fully nonlinear, degenerate elliptic Hamilton--Jacobi--Bellman (HJB) equation with a nontrivial Neumann boundary condition. Although these features have been studied separately, this work provides the first unified treatment combining them all. We establish that the optimal value function associated with the control problem is the unique viscosity solution of the HJB equation with a nontrivial Neumann boundary condition, and we present an illustrative example demonstrating the applicability of the framework.