Fall 2023 REFLECTED GENERALIZED BSDE WITH JUMPS UNDER STOCHASTIC CONDITIONS AND AN OBSTACLE PROBLEM FOR INTEGRAL-PARTIAL DIFFERENTIAL EQUATIONS WITH NONLINEAR NEUMANN BOUNDARY CONDITIONS
Mohammed Elhachemy, Mohamed El Otmani
J. Integral Equations Applications 35(3): 311-338 (Fall 2023). DOI: 10.1216/jie.2023.35.311

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.

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Mohammed Elhachemy. Mohamed El Otmani. "REFLECTED GENERALIZED BSDE WITH JUMPS UNDER STOCHASTIC CONDITIONS AND AN OBSTACLE PROBLEM FOR INTEGRAL-PARTIAL DIFFERENTIAL EQUATIONS WITH NONLINEAR NEUMANN BOUNDARY CONDITIONS." J. Integral Equations Applications 35 (3) 311 - 338, Fall 2023. https://doi.org/10.1216/jie.2023.35.311

Information

Received: 6 April 2023; Accepted: 11 September 2023; Published: Fall 2023
First available in Project Euclid: 25 October 2023

Digital Object Identifier: 10.1216/jie.2023.35.311

Subjects:
Primary: 35D40 , 35R09 , 60H05 , 60H10 , 60H30
Secondary: 60J60

Keywords: generalized BSDE with jumps , IPDE , nonlinear Neumann boundary conditions , obstacle , Reflected BSDE , stochastic Lipschitz , stochastic monotone , viscosity solution

Rights: Copyright © 2023 Rocky Mountain Mathematics Consortium

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Vol.35 • No. 3 • Fall 2023
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