A Kneser-type theorem for backward doubly stochastic differential equations

TL;DR

This paper investigates backward doubly stochastic differential equations with continuous coefficients, establishing comparison theorems, solution structures, and a Kneser-type theorem that characterizes the uniqueness or uncountability of solutions.

Contribution

It introduces a Kneser-type theorem for BDSDEs, revealing the conditions under which solutions are unique or uncountably many, expanding the theoretical understanding of these equations.

Findings

Existence of maximal solutions for BDSDEs with continuous coefficients

Comparison theorems for solutions of BDSDEs

A Kneser-type theorem indicating solution uniqueness or uncountability

Abstract

A class of backward doubly stochastic differential equations (BDSDEs in short) with continuous coefficients is studied. We give the comparison theorems, the existence of the maximal solution and the structure of solutions for BDSDEs with continuous coefficients. A Kneser-type theorem for BDSDEs is obtained. We show that there is either unique or uncountable solutions for this kind of BDSDEs.