Probabilistic max-plus schemes for solving Hamilton-Jacobi-Bellman equations

TL;DR

This paper introduces new probabilistic max-plus schemes for solving complex Hamilton-Jacobi-Bellman equations, improving monotonicity conditions and providing error estimates to enhance numerical solutions for diffusion control problems.

Contribution

The paper develops monotone probabilistic schemes under weaker assumptions and offers error estimates, advancing numerical methods for nonlinear HJB equations.

Findings

New schemes are monotone under weak assumptions

Error estimates for the proposed schemes

Enhanced numerical approach for diffusion control problems

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. In previous works (Akian, Fodjo, 2016 and 2017), we introduced a lower complexity probabilistic numerical algorithm for such equations by combining max-plus and numerical probabilistic approaches. The max-plus approach is in the spirit of the one of McEneaney, Kaise and Han (2011), and is based on the distributivity of monotone operators with respect to suprema. The numerical probabilistic approach is in the spirit of the one proposed by Fahim, Touzi and Warin (2011). A difficulty of the latter algorithm was in the critical constraints imposed on the Hamiltonian to ensure the monotonicity of the scheme, hence the convergence of the algorithm. Here, we present new probabilistic schemes which are monotone under rather weak assumptions, and show error estimates for these schemes. These estimates will be used in further works to study the probabilistic max-plus method.