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Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management

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  • Pham, Huyên

Abstract

We formulate and investigate a general stochastic control problem under a progressive enlargement of filtration. The global information is enlarged from a reference filtration and the knowledge of multiple random times together with associated marks when they occur. By working under a density hypothesis on the conditional joint distribution of the random times and marks, we prove a decomposition of the original stochastic control problem under the global filtration into classical stochastic control problems under the reference filtration, which is determined in a finite backward induction. Our method revisits and extends in particular stochastic control of diffusion processes with a finite number of jumps. This study is motivated by optimization problems arising in default risk management, and we provide applications of our decomposition result for the indifference pricing of defaultable claims, and the optimal investment under bilateral counterparty risk. The solutions are expressed in terms of BSDEs involving only Brownian filtration, and remarkably without jump terms coming from the default times and marks in the global filtration.

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  • Pham, Huyên, 2010. "Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1795-1820, August.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:9:p:1795-1820
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    References listed on IDEAS

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    1. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
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    Cited by:

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    2. Agostino Capponi & José Figueroa-López & Andrea Pascucci, 2015. "Dynamic credit investment in partially observed markets," Finance and Stochastics, Springer, vol. 19(4), pages 891-939, October.
    3. Idris Kharroubi & Thomas Lim, 2014. "Progressive Enlargement of Filtrations and Backward Stochastic Differential Equations with Jumps," Journal of Theoretical Probability, Springer, vol. 27(3), pages 683-724, September.
    4. Lijun Bo & Agostino Capponi, 2018. "Portfolio Choice with Market-Credit Risk Dependencies," Papers 1806.07175, arXiv.org.
    5. Erhan Bayraktar & Zhou Zhou, 2012. "On controller-stopper problems with jumps and their applications to indifference pricing of American options," Papers 1212.4894, arXiv.org, revised Nov 2013.
    6. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs with Jumps," CIRJE F-Series CIRJE-F-993, CIRJE, Faculty of Economics, University of Tokyo.
    7. Salvatore Federico & Paul Gassiat, 2014. "Viscosity Characterization of the Value Function of an Investment-Consumption Problem in Presence of an Illiquid Asset," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 966-991, March.
    8. Ludovic Tangpi & Shichun Wang, 2022. "Optimal Bubble Riding: A Mean Field Game with Varying Entry Times," Papers 2209.04001, arXiv.org, revised Jan 2024.
    9. Martin, Jessica & Villeneuve, Stéphane, 2021. "A Class of Explicit optimal contracts in the face of shutdown," TSE Working Papers 21-1183, Toulouse School of Economics (TSE), revised Apr 2022.
    10. Masaaki Fujii, 2014. "A Polynomial Scheme of Asymptotic Expansion for Backward SDEs and Option pricing," Papers 1405.0378, arXiv.org, revised Dec 2014.
    11. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs," CARF F-Series CARF-F-372, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    12. Jessica Martin & Stéphane Villeneuve, 2021. "A Class of Explicit optimal contracts in the face of shutdown," Working Papers hal-03124102, HAL.
    13. Alessandro Calvia & Emanuela Rosazza Gianin, 2019. "Risk measures and progressive enlargement of filtration: a BSDE approach," Papers 1904.13257, arXiv.org, revised Mar 2020.
    14. Bogdan Iftimie, 2023. "A robust investment-consumption optimization problem in a switching regime interest rate setting," Journal of Global Optimization, Springer, vol. 86(3), pages 713-739, July.
    15. Jessica Martin & Stéphane Villeneuve, 2023. "Risk-sharing and optimal contracts with large exogenous risks," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 46(1), pages 1-43, June.
    16. Giulia Di Nunno & Steffen Sjursen, 2013. "Information and optimal investment in defaultable assets," Papers 1312.6032, arXiv.org.
    17. Jessica Martin & St'ephane Villeneuve, 2021. "A Class of Explicit optimal contracts in the face of shutdown," Papers 2102.00001, arXiv.org.
    18. Song, Shiqi, 2016. "Drift operator in a viable expansion of information flow," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2297-2322.
    19. Masaaki Fujii & Akihiko Takahashi, 2015. "Asymptotic Expansion for Forward-Backward SDEs with Jumps," Papers 1510.03220, arXiv.org, revised Sep 2018.
    20. Masaaki Fujii, 2016. "A polynomial scheme of asymptotic expansion for backward SDEs and option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 16(3), pages 427-445, March.
    21. Christian Dehm & Thai Nguyen & Mitja Stadje, 2020. "Non-concave expected utility optimization with uncertain time horizon," Papers 2005.13831, arXiv.org, revised Oct 2021.
    22. Francesco Cordoni & Luca Di Persio & Luca Prezioso, 2019. "A lending scheme for a system of interconnected banks with probabilistic constraints of failure," Papers 1903.06042, arXiv.org, revised Oct 2019.
    23. Masaaki Fujii, 2014. "A Polynomial Scheme of Asymptotic Expansion for Backward SDEs and Option pricing," CARF F-Series CARF-F-343, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Dec 2014.
    24. Masaaki Fujii & Akihiko Takahashi, 2018. "Asymptotic Expansion for Forward-Backward SDEs with JumpsAsymptotic Expansion for Forward-Backward SDEs with Jumps (Forthcoming in Stochastics) (Revised version of F-372)," CARF F-Series CARF-F-445, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    25. Stéphane Crépey & Shiqi Song, 2014. "Counterparty risk and funding: Immersion and beyond," Working Papers hal-00989062, HAL.

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