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BSDEs with default jump

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  • Roxana Dumitrescu
  • Marie-Claire Quenez
  • Agn`es Sulem

Abstract

We study the properties of nonlinear Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale measure associated with a default jump with intensity process $(\lambda_t)$. We give a priori estimates for these equations and prove comparison and strict comparison theorems. These results are generalized to drivers involving a singular process. The special case of a $\lambda$-linear driver is studied, leading to a representation of the solution of the associated BSDE in terms of a conditional expectation and an adjoint exponential semi-martingale. We then apply these results to nonlinear pricing of European contingent claims in an imperfect financial market with a totally defaultable risky asset. The case of claims paying dividends is also studied via a singular process.

Suggested Citation

  • Roxana Dumitrescu & Marie-Claire Quenez & Agn`es Sulem, 2016. "BSDEs with default jump," Papers 1612.05681, arXiv.org, revised Sep 2017.
    • Handle: RePEc:arx:papers:1612.05681
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    References listed on IDEAS

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    1. Peter Bank & Dietmar Baum, 2004. "Hedging and Portfolio Optimization in Financial Markets with a Large Trader," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 1-18, January.
    2. Stefan Ankirchner & Christophette Blanchet-Scalliet & Anne Eyraud-Loisel, 2010. "CREDIT RISK PREMIA AND QUADRATIC BSDEs WITH A SINGLE JUMP," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(07), pages 1103-1129.
    3. Stefan Ankirchner & Christophette Blanchet-Scalliet & Anne Eyraud-Loisel, 2009. "Credit risk premia and quadratic BSDEs with a single jump," Papers 0907.1221, arXiv.org, revised Jun 2010.
    4. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    5. Roxana Dumitrescu & Marie-Claire Quenez & Agnès Sulem, 2015. "Game options in an imperfect market with default," Working Papers hal-01243603, HAL.
    6. Christophette Blanchet-Scalliet & Anne Eyraud-Loisel & Manuela Royer-Carenzi, 2010. "Hedging of Defaultable Contingent Claims using BSDE with uncertain time horizon," Post-Print hal-00341431, HAL.
    7. Quenez, Marie-Claire & Sulem, Agnès, 2013. "BSDEs with jumps, optimization and applications to dynamic risk measures," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 3328-3357.
    8. Ioannis Karatzas & (*), S. G. Kou, 1998. "Hedging American contingent claims with constrained portfolios," Finance and Stochastics, Springer, vol. 2(3), pages 215-258.
    9. Royer, Manuela, 2006. "Backward stochastic differential equations with jumps and related non-linear expectations," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1358-1376, October.
    10. Christophette Blanchet-Scalliet & Anne Eyraud-Loisel & Manuela Royer-Carenzi, 2010. "Hedging of Defaultable Contingent Claims using BSDE with uncertain time horizon," Post-Print hal-01107525, HAL.
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