  In this paper, we introduce a specific kind of doubly reflected Backward
Stochastic Differential Equations (in short DRBSDEs), defined on probability
spaces equipped with general filtration that is essentially non quasi-left
continuous, where the barriers are assumed to be predictable processes. We call
these equations predictable DRBSDEs. Under a general type of Mokobodzki's
condition, we show the existence of the solution (in consideration of the
driver's nature) through a Picard iteration method and a Banach fixed point
theorem. By using an appropriate generalization of It\^o's formula due to
Gal'chouk and Lenglart, we provide a suitable a priori estimates which
immediately implies the uniqueness of the solution.
