  We study the properties of nonlinear Backward Stochastic Differential
Equations (BSDEs) driven by a Brownian motion and a martingale measure
associated with a default jump with intensity process $(\lambda_t)$. We give a
priori estimates for these equations and prove comparison and strict comparison
theorems. These results are generalized to drivers involving a singular
process. The special case of a $\lambda$-linear driver is studied, leading to a
representation of the solution of the associated BSDE in terms of a conditional
expectation and an adjoint exponential semi-martingale. We then apply these
results to nonlinear pricing of European contingent claims in an imperfect
financial market with a totally defaultable risky asset. The case of claims
paying dividends is also studied via a singular process.
