Comparison Principle for Hamilton-Jacobi-Bellman Equations via a Bootstrapping Procedure

TL;DR

This paper establishes a comparison principle and existence results for Hamilton-Jacobi-Bellman equations with unbounded, discontinuous controls, broadening the scope of solutions in stochastic control and large deviations.

Contribution

It introduces a bootstrapping procedure to prove well-posedness of HJB equations with unbounded, discontinuous controls, extending previous theories.

Findings

Proved comparison principle for HJB equations without boundary conditions.

Established existence of viscosity solutions as value functions.

Applied results to two specific examples.

Abstract

We study the well-posedness of Hamilton-Jacobi-Bellman equations on subsets of $\mathbb{R}^d$ in a context without boundary conditions. The Hamiltonian is given as the supremum over two parts: an internal Hamiltonian depending on an external control variable and a cost functional penalizing the control. The key feature in this paper is that the control function can be unbounded and discontinuous. This way we can treat functionals that appear e.g. in the Donsker-Varadhan theory of large deviations for occupation time measures. To allow for this flexibility, we assume that the internal Hamiltonian and cost functional have controlled growth, and that they satisfy an equi-continuity estimate uniformly over compact sets in the space of controls. In addition to establishing the comparison principle for the Hamilton-Jacobi-Bellman equation, we also prove existence, the viscosity solution being the value function with exponentially discounted running costs. As an application, we verify the conditions on the internal Hamiltonian and cost functional in two examples.