  We study reflected backward stochastic differential equation (RBSDEs) on the
probability space equipped with a Brownian motion. The main novelty of the
paper lies in fact that we consider the following weak assumptions on the data:
barriers are optional of class (D) satisfying weak Mokobodzki's condition,
generator is continuous and non-increasing with respect to the value-variable
(no restriction on the growth) and Lipschitz continuous with respect to the
control-variable, and the terminal condition and the generator at zero are
supposed to be merely integrable. We prove that under these conditions on the
data there exists a solution to corresponding RBSDE. In the second part of the
paper, we apply the theory of RBSDEs to solve basic problems in Dynkin games
driven by nonlinear expectation based on the generator mentioned above. We
prove that the main component of a solution to RBSDE represents the value
process in corresponding extended nonlinear Dynkin game. Moreover, we provide
sufficient condition on the barriers guaranteeing the existence of the value
for nonlinear Dynkin games and the existence of a saddle point.
