## Stochastic representations for solutions to parabolic Dirichlet problems for nonlocal Bellman equations.(English)Zbl 1469.35212

Summary: 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.

### MSC:

 35R09 Integro-partial differential equations 35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations 35K65 Degenerate parabolic equations 49L20 Dynamic programming in optimal control and differential games 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H30 Applications of stochastic analysis (to PDEs, etc.) 93E20 Optimal stochastic control
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