Conditional Reflected Backward Stochastic Differential Equations with Two Barriers

TL;DR

This paper introduces doubly conditional reflected BSDEs with two barriers, establishing existence, uniqueness, and their relation to Dynkin games under partial information, with applications to investment stopping problems.

Contribution

It extends the theory of reflected BSDEs to a conditional setting with two barriers, linking solutions to Dynkin games with partial information.

Findings

Established existence and uniqueness under Mokobodski condition.

Linked the solution's conditional expectation to Dynkin game value functions.

Provided a weaker comparison theorem for these equations.

Abstract

In this paper, we study the doubly conditional reflected backward stochastic differential equations (BSDEs), where constraints are made on the conditional expectation of the first component of the solution with respect to a general subfiltration. With the help of the Skorokhod problem on a time-dependent interval and the Dynkin game in a general framework, we establish the existence and uniqueness result under the Mokobodski condition for the obstacles. The relation between the conditional expectation of the solution and the value function of a certain Dynkin game with partial information is obtained. As a by-product, we obtain a weaker version of the comparison theorem. Finally, we provide an application to the starting and stopping problem in reversible investments under partial information.