On the Quasi-linear Reflected Backward Stochastic Partial Differential Equations

TL;DR

This paper studies quasi-linear reflected backward stochastic partial differential equations, establishing existence, uniqueness, and well-posedness using variational, penalization, and continuity methods, and explores related stochastic calculus and optimal stopping connections.

Contribution

It introduces new methods for proving existence and uniqueness of solutions to quasi-linear RBSPDEs and extends the theory to include Itô formulas and comparison principles.

Findings

Proved existence and uniqueness of solutions for linear RBSPDEs with Laplacian coefficients.

Established well-posedness of general quasi-linear RBSPDEs.

Connected RBSPDE solutions with reflected backward stochastic differential equations and optimal stopping.

Abstract

This paper is concerned with the quasi-linear reflected backward stochastic partial differential equation (RBSPDE for short). Basing on the theory of backward stochastic partial differential equation and the parabolic capacity and potential, we first associate the RBSPDE to a variational problem, and via the penalization method, we prove the existence and uniqueness of the solution for linear RBSPDE with Lapalacian leading coefficients. With the continuity approach, we further obtain the well-posedness of general quasi-linear RBSPDEs. Related results, including It\^o formulas for backward stochastic partial differential equations with random measures, the comparison principle for solutions of RBSPDEs and the connections with reflected backward stochastic differential equations and optimal stopping problems, are addressed as well.