Reflected Solutions of Backward Doubly Stochastic Differential Equations

TL;DR

This paper investigates reflected backward doubly stochastic differential equations, establishing existence, uniqueness, and comparison theorems, with a novel proof approach for backward stochastic integrals.

Contribution

It introduces a new method for proving existence of solutions and extends the theory of reflected BDSDEs with comparison results.

Findings

Established existence and uniqueness of reflected BDSDE solutions

Developed a new proof technique for backward stochastic integrals

Proved a comparison theorem for reflected BDSDEs

Abstract

We study reflected solutions of one-dimensional backward doubly stochastic differential equations (BDSDEs in short). The "reflected" keeps the solution above a given stochastic process. We get the uniqueness and existence by penalization. For the existence of backward stochastic integral, our proof is different from [KKPPQ] slightly. We also obtain a comparison theorem for reflected BDSDEs.