  The paper is directly motivated by the pricing of vulnerable European and
American options in a general hazard process setup and a related study of the
corresponding pre-default backward stochastic differential equations (BSDE) and
pre-default reflected backward stochastic differential equations (RBSDE). We
work with a generic filtration $\FF$ for which the martingale representation
property is assumed to hold with respect to a square-integrable martingale $M$
and the goal of this work is of twofold. First, we aim to establish the
well-posedness results and comparison theorems for a generalized BSDE and a
reflected generalized BSDE with a continuous and nondecreasing driver $A$.
Second, we study extended penalization schemes for a generalized BSDE and a
reflected generalized BSDE in which we penalize against the driver in order to
obtain in the limit either a particular optimal stopping problem or a Dynkin
game in which the set of admissible exercise time is constrained to the right
support of the measure generated by $A$.
