Maximum Principle for Quasi-linear Reflected Backward SPDEs

TL;DR

This paper proves a maximum principle for quasi-linear reflected backward stochastic PDEs, establishing existence, uniqueness, and local behavior of solutions, and extending results to general domains.

Contribution

It introduces a maximum principle for RBSPDEs with non-zero boundary conditions and develops a stochastic De Giorgi iteration technique for general domains.

Findings

Maximum principle established for RBSPDEs on general domains

Existence and uniqueness of weak solutions proven

Local behavior of solutions analyzed

Abstract

This paper establishes a maximum principle for quasi-linear reflected backward stochastic partial differential equations (RBSPDEs for short). We prove the existence and uniqueness of the weak solution to RBSPDEs allowing for non-zero Dirichlet boundary conditions and, using a stochastic version of De Giorgi's iteration, establish the maximum principle for RBSPDEs on a general domain. The maximum principle for RBSPDEs on a bounded domain and the maximum principle for backward stochastic partial differential equations (BSPDEs for short) on a general domain can be obtained as byproducts. Finally, the local behavior of the weak solutions is considered.