We prove a stochastic representation formula for the viscosity solution of Dirichlet terminal-boundary value problem for a degenerate Hamilton–Jacobi–Bellman integro-partial differential equation in a bounded domain. We show that the unique viscosity solution is the value function of the associated stochastic optimal control problem. We also obtain the dynamic programming principle for the associated stochastic optimal control problem in a bounded domain.
"Stochastic representations for solutions to parabolic Dirichlet problems for nonlocal Bellman equations." Ann. Appl. Probab. 29 (6) 3271 - 3310, December 2019. https://doi.org/10.1214/19-AAP1473