Lp-solution of backward doubly stochastic differential equations

TL;DR

This paper advances the theory of backward doubly stochastic differential equations by establishing existence, uniqueness, and a priori estimates under weak assumptions, extending previous results to broader conditions.

Contribution

It introduces new stochastic calculus techniques and extends existence and uniqueness results for BDSDEs to cases with $L^p$ solutions and monotonicity conditions.

Findings

Established existence and uniqueness of solutions in $L^p$ spaces for BDSDEs.

Developed new technical tools in stochastic calculus related to BDSDEs.

Extended prior results to more general data assumptions.

Abstract

In this paper, our goal is solving backward doubly stochastic differential equation (BDSDE for short) under weak assumptions on the data. The first part of the paper is devoted to the development of some new technical aspects of stochastic calculus related to BDSDEs. Then we derive a priori estimates and prove existence and uniqueness of solutions, extending the results of Pardoux and Peng \cite{PP1} to the case where the solution is taked in $L^{p}, p>1$ and the monotonicity conditions are satisfied. This study is limited to deterministic terminal time.