Invariance times transfer properties

Abstract

Invariance times are stopping times $ \tau $ such that local martingales with respect to some reduced filtration and an equivalently changed probability measure, stopped before $ \tau $, are local martingales with respect to the original model filtration and probability measure. They arise naturally for modeling the default time of a dealer bank, in the mathematical finance context of counterparty credit risk. Assuming an invariance time endowed with an intensity and a positive Azéma supermartingale, this work establishes a dictionary relating the semimartingale calculi in the original and reduced stochastic bases, regarding conditional expectations, martingales, stochastic integrals, random measure stochastic integrals, martingale representation properties, semimartingale characteristics, Markov properties, transition semigroups and infinitesimal generators, and solutions of backward stochastic differential equations.

Keywords

Progressive enlargement of filtration / Invariance time / Semimartingale calculus / Markov process / Backward stochastic differential equation / Counterparty risk / Credit risk

Acknowledgements

This research has benefited from the support of the Chair Capital Markets Tomorrow: Modeling and Computational Issues under the aegis of the Institut Europlace de Finance, a joint initiative of Laboratoire de Probabilités, Statistique et Modélisation (LPSM)/Université Paris Cité and Crédit Agricole CIB. The author is grateful to Shiqi Song for his contributions to a preliminary version of this work and to Martin Schweizer and Monique Jeanblanc for precious comments.

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