Strong snell envelopes and RBSDEs with regulated trajectories when the barrier is a semimartingale

Abstract

We consider reflected backward stochastic different equations (RBSDEs) with lower optional barrier and so-called regulated trajectories, i.e trajectories with left and right finite limits (làdlàg), which is assumed to be right upper-semicontinuous semimartingale. By using some tools from the general theory of processes such as Mertens decomposition of optional strong supermartingales, we provide the link between the decomposition of the barrier and that of its strong Snell envelope which is the solution of RBSDE.