Doubly reflected BSDEs with default time under stochastic Lipschitz coefficients: Filtration links and generalized Dynkin games

Abstract

We study doubly reflected backward stochastic differential equations (DRBSDEs) on a random horizon $T\wedge \tau$ generated by a default time $\tau$ in a progressively enlarged filtration. We work within the change-of-measure framework previously developed for progressive enlargement, under which the reference Brownian motion stopped at the default time remains a Brownian motion in the enlarged filtration. The DRBSDEs are driven by stochastic Lipschitz generators and involve two RCLL completely separated barriers. We first establish existence and uniqueness of solutions in a weighted square-integrable framework, allowing for an additional martingale component orthogonal to the stopped Brownian martingale. We then investigate the relationship between DRBSDEs formulated in the enlarged filtration and related equations in the reference Brownian filtration. Finally, we formulate a nonlinear Dynkin game on the random horizon $T\wedge \tau$ and characterize its value process in terms of the solution of the associated DRBSDE. Under suitable regularity assumptions, we also identify a saddle point through the first hitting times of the lower and upper barriers.