The obstacle problem for semilinear parabolic partial integro-differential equations.

Summary We give a probabilistic interpretation for the weak Sobolev solution of obstacle problem for semilinear parabolic partial integro-differential equations (PIDE). The results of Léandre [29] about the homeomorphic property for the solution of SDE with jumps are used to construct random test functions for the variational equation for such PIDE. This yields to the natural connection with the associated Reflected Backward Stochastic Differential Equations with jumps (RBSDE), namely the Feynman Kac's formula for the solution of the PIDE. MSC: 60H15. 60G46. 35R60 Keyword: Reflected backward stochastic differential equation, partial parabolic integro-differential equation, jump diffusion process, obstacle problem, stochastic flow, flow of diffeo-morphism.