Doubly Reflected BSDEs with Integrable Parameters and Related Dynkin Games

TL;DR

This paper investigates doubly reflected BSDEs with integrable parameters, establishing a unique solution and linking it to the value process of a related Dynkin game under nonlinear g-evaluation.

Contribution

It constructs a unique solution for doubly reflected BSDEs with separated obstacles and connects it to the Dynkin game value under g-evaluation.

Findings

Unique solution for doubly reflected BSDEs with separated obstacles.

The process Y represents the Dynkin game value under g-evaluation.

First hitting times of Y with obstacles form saddle points of the game.

Abstract

We study a doubly reflected backward stochastic differential equation (BSDE) with integrable parameters and the related Dynkin game. When the lower obstacle $L$ and the upper obstacle $U$ of the equation are completely separated, we construct a unique solution of the doubly reflected BSDE by pasting local solutions and show that the $Y-$component of the unique solution represents the value process of the corresponding Dynkin game under $g-$evaluation, a nonlinear expectation induced by BSDEs with the same generator $g$ as the doubly reflected BSDE concerned. In particular, the first time when process $Y $ meets $L$ and the first time when process $Y $ meets $U$ form a saddle point of the Dynkin game.