BSDEs of Counterparty Risk and Invariant Times

Stéphane Crépey (Evry University)
Thursday, November 6, 2014 - 4:00pm
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

This work is motivated by the need to generalize the classical credit risk reduced-form modeling approach for counterparty risk applications. We relax the basic immersion conditions of the classical approach by modeling the default time as an invariant time, such that local martingales with respect to a reduced filtration and a possibly changed probability measure, once stopped right before that time, stay local martingales with respect to the original model filtration and probability measure. Specifically, we study a BSDE with random terminal time that appears in the modeling of counterparty risk in finance. We proceed by reduction of the original BSDE into a simpler BSDE posed with respect to a subfiltration and a changed probability measure. This is done under a relaxation of the classical immersion hypothesis, stated in terms of the changed probability measure, of which we determine the Radon-Nikodym derivative. We provide an Azema supermartingale characterization of invariant times and use it for establishing the equivalence between the original and the reduced BSDE. This allows proving well-posedness of the original nonstandard BSDE by well-posedness of the reduced BSDE, which holds under classical assumptions.

This is a joint work with Shiqi Song.