Reflected backward stochastic differential equations with two barriers and Dynkin games under Knightian uncertainty

This paper is concerned with a class of reflected backward stochastic differential equations (RBSDEs in short) with two barriers. The first purpose of the paper is to establish existence and uniqueness results of adapted solutions for such RBSDEs. Most of existing results on adapted solutions for RBSDEs with two barriers are heavily based on either the Mokobodski condition or other restrictive regularity conditions. In this paper, the two barriers are modeled by stochastic differential equations with coefficients satisfying the local Lipschitz condition and the linear growth condition, which enables us to weaken the regularity conditions on the boundary processes. Existence is proved by a penalization scheme together with a comparison theorem under the Lipschitz condition on the coefficients of RBSDEs. As an application, it is proved that the initial value of an RBSDE with two barriers coincides with the value function of a certain Dynkin game under Knightian uncertainty.