Some regularity and convergence results for parabolic Hamilton-Jacobi-Bellman equations in bounded domains

TL;DR

This paper investigates the approximation and convergence of numerical schemes for parabolic Hamilton-Jacobi-Bellman equations in bounded domains, establishing error bounds and solution regularity under certain boundary conditions.

Contribution

It introduces a novel approach to prove existence of solutions via a switching system and derives boundary error estimates for monotone schemes.

Findings

Existence of viscosity solutions via a new sub- and supersolution construction.

Error bounds for finite difference and semi-Lagrangian schemes near boundaries.

Convergence results comparable to those in unbounded domains.

Abstract

We study the approximation of parabolic Hamilton-Jacobi-Bellman (HJB) equations in bounded domains with strong Dirichlet boundary conditions. We work under the assumption of the existence of a sufficiently regular barrier function for the problem to obtain well-posedness and regularity of a related switching system and the convergence of its components to the HJB equation. In particular, we show existence of a viscosity solution to the switching system by a novel construction of sub- and supersolutions and application of Perron's method. Error bounds for monotone schemes for the HJB equation are then derived from estimates near the boundary, where the standard regularisation procedure for viscosity solutions is not applicable, and are found to be of the same order as known results for the whole space. We deduce error bounds for some common finite difference and truncated semi-Lagrangian schemes.