Maximum Principle for Quasi-linear Backward Stochastic Partial Differential Equations

TL;DR

This paper establishes maximum principles and regularity results for quasi-linear backward stochastic partial differential equations of parabolic type, providing existence, uniqueness, and maximum estimates for weak solutions.

Contribution

It proves the existence and uniqueness of weak solutions and develops maximum principles using De Giorgi iteration for quasi-linear BSPDEs.

Findings

Existence and uniqueness of weak solutions

Maximum estimates and global maximum principle

Local maximum principle for stochastic parabolic De Giorgi class

Abstract

In this paper we are concerned with the maximum principle for quasi-linear backward stochastic partial differential equations (BSPDEs for short) of parabolic type. We first prove the existence and uniqueness of the weak solution to quasi-linear BSPDE with the null Dirichlet condition on the lateral boundary. Then using the De Giorgi iteration scheme, we establish the maximum estimates and the global maximum principle for quasi-linear BSPDEs. To study the local regularity of weak solutions, we also prove a local maximum principle for the backward stochastic parabolic De Giorgi class.