  We propose a parallel algorithm for the numerical solution of a class of
second order semi-linear equations coming from stochastic optimal control
problems, by means of a dynamic domain decomposition technique. The new method
is an extension of the patchy domain decomposition method presented in a
previous work for first order Hamilton-Jacobi-Bellman equations related to
deterministic optimal control problems. The semi-Lagrangian scheme underlying
the original method is modified in order to deal with (possibly degenerate)
diffusion, by approximating the stochastic optimal control problem associated
to the equation via discrete time Markov chains. We show that under suitable
conditions on the discretization parameters and for sufficiently small values
of the diffusion coefficient, the parallel computation on the proposed dynamic
decomposition is faster than that on a static decomposition. To this end, we
combine the parallelization with some well known techniques in the context of
fast-marching-like methods for first order Hamilton-Jacobi equations. Several
numerical tests in dimension two are presented, in order to show the features
of the proposed method.
