A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance

TL;DR

This paper introduces a penalty method for efficiently solving Hamilton-Jacobi-Bellman equations numerically, especially in finance, by approximating nonlinear systems with iterative schemes that converge finitely.

Contribution

The paper proposes a novel penalty method that simplifies solving nonlinear HJB equations, ensuring convergence and applicability in financial models.

Findings

Method converges in finitely many steps

Applicable to a broad class of HJB equations

Demonstrated effectiveness with financial examples

Abstract

We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately, many intuitive (e.g. finite difference based) discretisations can be shown to converge. However, especially when using fully implicit time stepping schemes with their desirable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to first order in the penalty parameter, and we show that an iterative scheme can be used to solve the penalised discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results.