Stochastic optimal control problems for a class of reflected diffusion with Poisson jumps in a half_space are considered. The nonlinear Nisio’s semigroup for such optimal control problems was constructed. A Hamilton_Jacobi_Bellman equation with the Neumann boundary condition associated with this semigroup was obtained. Then, viscosity solutions of this equation were defined and discussed, and various uniqueness of this equation was also considered. Finally, the value function was such optimal control problems is shown to be a viscosity solution of this equation.
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