Reflected generalized BDSDEs driven by non-homogeneous Lévy processes and obstacle problems for stochastic integro-PDEs with nonlinear Neumann boundary conditions

TL;DR

This work analyzes reflected generalized backward doubly SDEs driven by a non-homogeneous Lévy process, proving existence and uniqueness under stochastic monotone and Lipschitz-type conditions on the generators. It develops a comprehensive RGBDSDE-NL framework and a Picard–Yosida approach, alongside an extended Itô formula, to obtain a probabilistic representation for viscosity solutions of obstacle problems for stochastic integro-partial differential equations with nonlinear Neumann boundary conditions. The authors then build a Markovian, forward–backward structure linking RGBDSDE-NL to stochastic IPDEs via the Doss–Sussman transform, establishing that the solution field $u(t,x)$ is a stochastic viscosity solution, and extend this to obstacle SIPDE-NBCs through reflected GBDSDEs. The results extend the Brownian and homogeneous Lévy settings to non-homogeneous Lévy noise with finite-jump-size structure, providing a rigorous foundation for probabilistic representations of stochastic IPDEs with nonlinear Neumann boundaries in obstacle problems and their applications.

Abstract

We consider reflected generalized backward doubly stochastic differential equations driven by a non-homogeneous Lévy process. Under stochastic conditions on the coefficients, we prove the existence and uniqueness of a solution. Furthermore, we apply these results to obtain a probabilistic representation for the viscosity solutions of an obstacle problem governed by stochastic integro-partial differential equations with a nonlinear Neumann boundary condition.

Theorems & Definitions (45)

• Definition 1
• Remark 2
• Lemma 3
• Definition 4
• Remark 5
• Lemma 6
• Corollary 7
• proof
• Proposition 8
• proof