The identification problem for BSDEs driven by possibly non-quasi-left-continuous random measures

In this paper, we focus on the so-called identification problem for a BSDE driven by a continuous local martingale and a possibly non-quasi-left-continuous random measure. Supposing that a solution (Y,Z,U) of a BSDE is such that Yt = v(t,Xt), where X is an underlying process and v is a deterministic function, solving the identification problem consists in determining Z and U in terms of v. We study the over-mentioned identification problem under various sets of assumptions and we provide a family of examples including the case when X is a non-semimartingale jump process solution of an SDE with singular coefficients.