Progressive enlargement of filtrations and Backward SDEs with jumps

Idris Kharroubi 1 Thomas Lim 2
1 CEREMADE
CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
Abstract : This work deals with backward stochastic differential equation (BSDE) 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 appli-cations, 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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Journal of Theoretical Probability, Sprnger, 2014, pp.683-724. 〈10.1007/s10959-012-0428-1〉
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Idris Kharroubi, Thomas Lim. Progressive enlargement of filtrations and Backward SDEs with jumps . Journal of Theoretical Probability, Sprnger, 2014, pp.683-724. 〈10.1007/s10959-012-0428-1〉. 〈hal-01103709〉

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