BSDEs and Reflected BSDEs with default time and stochastic Lipschitz drivers: Theory and some financial applications

In this paper, we investigate nonlinear backward and reflected backward stochastic differential equations (BSDEs and RBSDEs) in a defaultable setting with $ \gamma $-stochastic Lipschitz drivers, where $ (\gamma_t) $ denotes the intensity process associated with the compensated martingale linked to the default time. We establish the existence and uniqueness of solutions for BSDEs using the Picard iteration method and for RBSDEs through a penalization approach combined with classical results from optimal control theory. As an application, we address the fair pricing of European and American options within a general financial market model that includes default risk, considering both (linear) perfect and (nonlinear) imperfect market conditions.