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

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.