  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.
