Stochastic solutions to Hamilton-Jacobi-Bellman Dirichlet problems (2404.17236v3)
Abstract: 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 that its upper and lower envelopes are viscosity sub- and supersolutions, respectively. Moreover, 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 a strong Feller selection, which propagates continuity to the stochastic solution. By means of an example, we also discuss the necessity of certain ellipticity assumptions for the continuity of the stochastic solution.
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