We present results on Reflected Backward Stochastic Differential Equations (RBSDE for short) on Brownian filtration with two optional barriers satisfying some separation condition.

It has been widely recognized that RBSDEs provide a useful framework for studying problems in many fields, such as financial mathematics, stochastic optimal control and partial differential equations (e.g. optimal stopping problem, Dynkin games, stopping and control games, switching problem, PDEs with singular data, homogenization, boundary problems, regularity problems, numerical schemes etc.).

The theory of RBSDEs is well studied for càdlàg barriers. However, there is only a few papers concerning BSDEs with non-càdlàg barriers. It is caused mostly by the following: the main component of the solution does not have to be a càdlàg process, the minimality condition (guaranteeing uniqueness) is complicated and unintuitive, and the basic proof technique of càdlàg case, i.e. penalization method, does not apply to non-càdlàg case.

We will present the results on the existence and uniqueness of a solution to RBSDEs with two optional barriers. The generator is assumed to be non-increasing with respect to the value variable (with no restrictions on the growth) and Lipschitz continuous, with sublinear growth, with respect to the control variable. Data are in Lp for some p≥1. We also present some results concerning methods of approximation of the solution via modified penalization scheme, and present the link between solutions of RBSDEs with optional barriers and the value processes for generalized nonlinear Dynkin games.

The results were obtained in cooperation with Tomasz Klimsiak and Leszek Słomiński.