  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.
