  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 of viscosity solutions 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 connect the strong
Markov selection to a Feller selection, which entails the continuity of the
stochastic solution.
