Doubly Reflected BSDEs with Integrable Parameters and Related Dynkin Games

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 $\tau_*$ when process $Y$ meets $L$ and the first time $\gamma_*$ when process $Y$ meets $U$ form a saddle point of the Dynkin game.

Keywords: BSDEs, Reflected BSDEs, Doubly reflected BSDEs, g-evaluation/expectation, Penalization, Optimal stopping problems, Pasting local solutions, Dynkin games, Saddle points.