Generalized BSDE and reflected BSDE with random time horizon

Abstract

Motivated by structural, reduced-form and hybrid models of the third party and counterparty credit risk, we study a generalized backward stochastic differential equations (BSDE) up to a random time horizon ϑ, which is not a stopping time with respect to a reference filtration. In contrast to the existing literature in the area of credit risk modeling, we do not impose specific assumptions on the random time ϑ and we study the existence of solutions to BSDE and reflected BSDE with a random time horizon through the method of reduction. For this purpose, we also examine BSDE and reflected BSDE with a làdlàg driver where the driver is allowed to have a finite number of jumps overlapping with jumps of the martingale part. Theoretical results are illustrated by particular instances of a random time and explicit BSDEs in either the Brownian or Brownian-Poisson filtration.