Stochastic solutions to Hamilton-Jacobi-Bellman Dirichlet problems

We consider a nonlinear Dirichlet problem on a bounded domain whose Hamiltonian is given by a Hamilton-Jacobi-Bellman operator with merely continuous and bounded coefficients. The objective of this paper is to study the existence question from a stochastic point of view. Using a relaxed control framework, we define a candidate for a viscosity solution. We prove that this stochastic solution satisfies viscosity sub- and supersolution properties and we establish a strong Markov selection principle, which shows that the stochastic solution can be realized through a strong Markov family. Building on the selection principle, we investigate regularity properties of the stochastic solution. For certain elliptic cases, we show that the strong Markov selection is even a Feller selection, which entails the continuity of the stochastic solution.