A Viscosity Approach to a Stochastic Control Problem on a Bounded Domain

TL;DR

This paper investigates a stochastic control problem within a bounded domain, demonstrating that the value function is a viscosity solution to the associated Hamilton-Jacobi-Bellman equation, with proofs of continuity, dynamic programming, and uniqueness.

Contribution

It introduces a viscosity solution framework for the stochastic control problem on bounded domains, establishing continuity, the dynamic programming principle, and solution uniqueness.

Findings

Value function is jointly continuous.

Value function satisfies the Dynamic Programming Principle.

Uniqueness of the viscosity solution is proven.

Abstract

We study a stochastic control problem on a bounded domain, which arises from a continuous-time optimal management model. Via the corresponding Hamilton-Jacobi-Bellman equation the value function is shown to be jointly continuous and to satisfy the Dynamic Programming Principle. These properties directly lead to the conclusion that the value function is a viscosity solution to the Hamilton-Jacobi-Bellman equation. Uniqueness of the solution is then also established.