Reflected BSDEs when the obstacle is not right-continuous in a general filtration

Abstract

We prove existence and uniqueness of the reflected backward stochastic differential equation's (RBSDE) solution with a lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous in a filtration that supports a Brownian motion $W$ and an independent Poisson random measure $\pi$. The result is established by using some tools from the general theory of processes such as Mertens decomposition of optional strong (but not necessarily right continuous) supermartingales and some tools from optimal stopping theory, as well as an appropriate generalization of It\^{o}'s formula due to Gal'chouk and Lenglart. Two applications on dynamic risk measure and on optimal stopping will be given.