  In a previous work (Akian, Fodjo, 2016), we introduced a lower complexity
probabilistic max-plus numerical method for solving 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. This method was based on the idempotent expansion
properties obtained by McEneaney, Kaise and Han (2011) and on the numerical
probabilistic method proposed by Fahim, Touzi and Warin (2011) for solving some
fully nonlinear parabolic partial differential equations. A difficulty of the
algorithm of Fahim, Touzi and Warin is in the critical constraints imposed on
the Hamiltonian to ensure the monotonicity of the scheme, hence the convergence
of the algorithm. Here, we propose a new "probabilistic scheme" which is
monotone under rather weak assumptions, including the case of strongly elliptic
PDE with bounded coefficients. This allows us to apply our probabilistic
max-plus method in more general situations. We illustrate this on the
evaluation of the superhedging price of an option under uncertain correlation
model with several underlying stocks and changing sign cross gamma, and
consider in particular the case of 5 stocks leading to a PDE in dimension 5.
