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Well-posedness and penalization schemes for generalized BSDEs and reflected generalized BSDEs

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  • Libo Li
  • Ruyi Liu
  • Marek Rutkowski

Abstract

The paper is directly motivated by the pricing of vulnerable European and American options in a general hazard process setup and a related study of the corresponding pre-default backward stochastic differential equations (BSDE) and pre-default reflected backward stochastic differential equations (RBSDE). We work with a generic filtration $\FF$ for which the martingale representation property is assumed to hold with respect to a square-integrable martingale $M$ and the goal of this work is of twofold. First, we aim to establish the well-posedness results and comparison theorems for a generalized BSDE and a reflected generalized BSDE with a continuous and nondecreasing driver $A$. Second, we study extended penalization schemes for a generalized BSDE and a reflected generalized BSDE in which we penalize against the driver in order to obtain in the limit either a particular optimal stopping problem or a Dynkin game in which the set of admissible exercise time is constrained to the right support of the measure generated by $A$.

Suggested Citation

  • Libo Li & Ruyi Liu & Marek Rutkowski, 2022. "Well-posedness and penalization schemes for generalized BSDEs and reflected generalized BSDEs," Papers 2212.12854, arXiv.org.
    • Handle: RePEc:arx:papers:2212.12854
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    References listed on IDEAS

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    1. 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.
    2. Tahir Choulli & Catherine Daveloose & Michèle Vanmaele, 2020. "A martingale representation theorem and valuation of defaultable securities," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1527-1564, October.
    3. Miryana Grigorova & Peter Imkeller & Elias Offen & Youssef Ouknine & Marie-Claire Quenez, 2017. "Reflected BSDEs when the obstacle is not right-continuous and optimal stopping," Post-Print hal-01141801, HAL.
    4. Miryana Grigorova & Peter Imkeller & Elias Offen & Youssef Ouknine & Marie-Claire Quenez, 2015. "Reflected BSDEs when the obstacle is not right-continuous and optimal stopping," Papers 1504.06094, arXiv.org, revised May 2017.
    5. Libo Li & Ruyi Liu & Marek Rutkowski, 2022. "Vulnerable European and American Options in a Market Model with Optional Hazard Process," Papers 2212.12860, arXiv.org.
    6. Essaky, E.H. & Hassani, M. & Ouknine, Y., 2015. "Stochastic quadratic BSDE with two RCLL obstacles," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2147-2189.
    7. Stéphane Crépey & Shiqi Song, 2016. "Counterparty risk and funding: immersion and beyond," Finance and Stochastics, Springer, vol. 20(4), pages 901-930, October.
    8. Miryana Grigorova & Marie-Claire Quenez & Agnès Sulem, 2020. "European options in a non-linear incomplete market model with default," Post-Print hal-02025833, HAL.
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    Cited by:

    1. Libo Li & Ruyi Liu & Marek Rutkowski, 2022. "Vulnerable European and American Options in a Market Model with Optional Hazard Process," Papers 2212.12860, arXiv.org.

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