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BSDEs with Default Jump

  • Roxana Dumitrescu
  • Miryana Grigorova
  • Marie-Claire Quenez
  • Agnès Sulem
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)

Abstract

We study (nonlinear) Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale attached to a default jump with intensity process λ = (λt). The driver of the BSDEs can be of a generalized form involving a singular optional finite variation process. In particular, we provide a comparison theorem and a strict comparison theorem. In the special case of a generalized λ-linear driver, we show an explicit representation of the solution, involving conditional expectation and an adjoint exponential semimartingale; for this representation, we distinguish the case where the singular component of the driver is predictable and the case where it is only optional. We apply our results to the problem of (nonlinear) pricing of European contingent claims in an imperfect market with default. We also study the case of claims generating intermediate cashflows, in particular at the default time, which are modeled by a singular optional process. We give an illustrating example when the seller of the European option is a large investor whose portfolio strategy can influence the probability of default.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Roxana Dumitrescu
    • 1
  • Miryana Grigorova
    • 2
  • Marie-Claire Quenez
    • 3
  • Agnès Sulem
    • 4
    • 5
  1. 1.Department of MathematicsKing’s College LondonLondonUK
  2. 2.Centre for Mathematical EconomicsUniversity BielefeldBielefeldGermany
  3. 3.Laboratoire de probabilités, statistiques et modélisations (CNRS/Sorbonne Université/Université Paris Diderot)Université Paris 7ParisFrance
  4. 4.INRIA ParisParis Cedex 12France
  5. 5.Université Paris-EstChamps-sur-MarneFrance

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