Backward doubly stochastic differential equations with or without reflection under weak conditions

TL;DR

This paper investigates the existence, uniqueness, and comparison principles for backward doubly stochastic differential equations (BDSDEs) with or without reflection under various growth conditions on the generator, broadening theoretical understanding.

Contribution

It establishes well-posedness results for BDSDEs with general, linear, and quadratic growth in the generator under weak conditions, including reflection cases.

Findings

Proved existence and uniqueness for BDSDEs with general growth in y and linear in z.

Established comparison principles for reflected and non-reflected BDSDEs.

Demonstrated existence of maximal solutions under quadratic growth conditions.

Abstract

In this paper, we study the well-posedness of backward doubly stochastic differential equations (BDSDEs), both with and without reflection, under weak conditions. First, when the generator $f$ is of general growth in $y$ and linear growth in $z$, we establish the existence, uniqueness, comparison principle, and the existence of maximal solutions for BDSDEs, with or without reflection. Second, under the assumption that $f$ is of linear growth in $y$ and quadratic growth in $z$, and that the terminal value is bounded, we prove the existence, uniqueness, and comparison principle for reflected and non-reflected BDSDEs. Finally, when the generator $f$ is of general growth in $y$ and quadratic growth in $z$, again with a bounded terminal value, we prove the existence of maximal solutions for BDSDEs in both the reflected and non-reflected situations.