Combined Fixed Point and Policy Iteration for Hamilton--Jacobi--Bellman Equations in Finance

Abstract

Implicit methods for Hamilton--Jacobi--Bellman (HJB) partial differential equations give rise to highly nonlinear discretized algebraic equations. The classic policy iteration approach may not be efficient in many circumstances. In this article, we derive sufficient conditions to ensure convergence of a combined fixed point policy iteration scheme for the solution of discretized equations. Numerical examples are included for a singular stochastic control problem arising in insurance (a guaranteed minimum withdrawal benefit), where the underlying risky asset follows a jump diffusion, and an American option assuming a regime switching process.