Existence and uniqueness for reflected BSDE with multivariate point process and right upper semicontinuous obstacle

Abstract

Abstract In a noise driven by a multivariate point process μ with predictable compensator ν, we prove existence and uniqueness of the reflected backward stochastic differential equation’s solution with a lower obstacle ( ξ t ) t ∈ [ 0 , T ] {(\xi_{t})_{t\in[0,T]}} which is assumed to be a right upper-semicontinuous, but not necessarily right-continuous process, and a Lipschitz driver f. The result is established by using the Mertens decomposition of optional strong (but not necessarily right continuous) super-martingales, an appropriate generalization of Itô’s formula due to Gal’chouk and Lenglart and some tools from optimal stopping theory. A comparison theorem for this type of equations is given.