Stochastic differential equations with respect to optional semimartingales and two reflecting regulated barriers

In this work, we introduce a new Skorokhod problem with two reflecting barriers, where the trajectories of the driving process and the barriers are right- and left-limited. We show that this problem has a unique explicit solution. Then, we apply our results to study the existence and uniqueness of solutions to reflected stochastic differential equations driven by optional semimartingales. The study is carried out on a probability space that does not necessarily satisfy the usual conditions.