REFLECTED GENERALIZED BSDE WITH JUMPS UNDER STOCHASTIC CONDITIONS AND AN OBSTACLE PROBLEM FOR INTEGRAL-PARTIAL DIFFERENTIAL EQUATIONS WITH NONLINEAR NEUMANN BOUNDARY CONDITIONS

Abstract

By a probabilistic approach, we look at an obstacle problem with nonlinear Neumann boundary conditions for parabolic semilinear integral-partial differential equations. We prove the existence of a continuous viscosity solution of this problem. The nonlinear part of the equation and the Neumann condition satisfy the stochastic monotonicity condition on the solution variable. Furthermore, the nonlinear part is stochastic Lipschitz on the parts that depend on the gradient and the integral of the solution. It should be noted that the existence of the viscosity solution for this problem has recently been investigated using a standard monotonicity and Lipschitz conditions. We show that the solution of the related reflected generalized backward stochastic differential equations with jumps exists and is unique when the barrier is right continuous left limited (rcll) and the generators satisfy stochastic monotonicity and Lipschitz conditions. In this case, we get a comparison result.