BSDEs with default jump

TL;DR

This paper analyzes nonlinear BSDEs driven by Brownian motion and default jump martingales, providing estimates, comparison theorems, and applications to pricing in defaultable markets.

Contribution

It introduces new properties, estimates, and comparison results for BSDEs with default jumps, including a representation for solutions with linear drivers and applications to financial pricing.

Findings

Established a priori estimates for BSDEs with default jumps.

Proved comparison and strict comparison theorems for these BSDEs.

Applied results to nonlinear pricing of European claims in defaultable markets.

Abstract

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.