Progressive Enlargement of Filtrations and Backward Stochastic Differential Equations with Jumps |
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Authors: | Idris Kharroubi Thomas Lim |
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Institution: | 1. CEREMADE, CNRS UMR 7534, Université Paris Dauphine, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France 2. Laboratoire d’Analyse et Probabilités, Université d’Evry and ENSIIE, 1, Square de la Résistance, 91025, Evry Cedex, France
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Abstract: | This work deals with backward stochastic differential equations (BSDEs for short) with random marked jumps, and their applications to default risk. We show that these BSDEs are linked with Brownian BSDEs through the decomposition of processes with respect to the progressive enlargement of filtrations. We prove that the equations have solutions if the associated Brownian BSDEs have solutions. We also provide a uniqueness theorem for BSDEs with jumps by giving a comparison theorem based on the comparison for Brownian BSDEs. We give in particular some results for quadratic BSDEs. As applications, we study the pricing and the hedging of a European option in a market with a single jump, and the utility maximization problem in an incomplete market with a finite number of jumps. |
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