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

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.