Progressive enlargement of filtrations and Backward SDEs with jumps

Idris Kharroubi (1) , Thomas Lim (2, 3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 LAP - Laboratoire Analyse et Probabilités
3 ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise

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 show 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 BDSEs. As applications, we study the pricing and the hedging of a European option in a complete market with a single jump, and the utility maximization problem in an incomplete market with a finite number of jumps.

Keywords: Backward SDE; quadratic BSDE; multiple random marked times; progressive enlargement of filtrations; decomposition in the reference filtration; exponential utility; exponential utility.

Domains: Optimization and Control [math.OC]; Probability [math.PR]

Idris Kharroubi, Thomas Lim. Progressive enlargement of filtrations and Backward SDEs with jumps. 2011. ⟨hal-00555787v2⟩