A probabilistic max-plus numerical method for solving stochastic control problems

TL;DR

This paper introduces a probabilistic numerical method combining idempotent expansion and probabilistic techniques to efficiently solve complex stochastic control problems modeled by Hamilton-Jacobi-Bellman equations.

Contribution

It develops a lower complexity probabilistic algorithm for fully nonlinear HJB equations with switching and continuum controls, improving computational efficiency.

Findings

Successfully applied to option pricing and hedging example

Demonstrates reduced computational complexity

Validates effectiveness through numerical tests

Abstract

We consider fully nonlinear Hamilton-Jacobi-Bellman equations associated to diffusion control problems involving a finite set-valued (or switching) control and possibly a continuum-valued control. We construct a lower complexity probabilistic numerical algorithm by combining the idempotent expansion properties obtained by McEneaney, Kaise and Han (2011) for solving such problems with a numerical probabilistic method such as the one proposed by Fahim, Touzi and Warin (2011) for solving some fully nonlinear parabolic partial differential equations. Numerical tests on a small example of pricing and hedging an option are presented.