Numerical solution of a semilinear parabolic degenerate Hamilton-Jacobi-Bellman equation with singularity

TL;DR

This paper develops numerical schemes for solving a degenerate semilinear parabolic Hamilton-Jacobi-Bellman equation with singular initial conditions, related to stochastic control with fuel constraints, and proves their convergence.

Contribution

It introduces a variable transformation to handle singularities and constructs convergent explicit and implicit numerical schemes for the transformed HJB equation.

Findings

Successfully transforms the singular HJB into a regular form

Constructs explicit and implicit numerical schemes

Proves convergence of the proposed schemes

Abstract

We consider a semilinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity which is related to a stochastic control problem with fuel constraint. The fuel constraint translates into a singular initial condition for the HJB equation. We first propose a transformation based on a change of variables that gives rise to an equivalent HJB equation with nonsingular initial condition but irregular coefficients. We then construct explicit and implicit numerical schemes for solving the transformed HJB equation and prove their convergences by establishing an extension to the result of Barles and Souganidis (1991).