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Representation for martingales living after a random time with applications

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  • Tahir Choulli
  • Ferdoos Alharbi

Abstract

Our financial setting consists of a market model with two flows of information. The smallest flow F is the "public" flow of information which is available to all agents, while the larger flow G has additional information about the occurrence of a random time T. This random time can model the default time in credit risk or death time in life insurance. Hence the filtration G is the progressive enlargement of F with T. In this framework, under some mild assumptions on the pair (F, T), we describe explicitly how G-local martingales can be represented in terms of F-local martingale and parameters of T. This representation complements Choulli, Daveloose and Vanmaele \cite{ChoulliDavelooseVanmaele} to the case when martingales live "after T". The application of these results to the explicit parametrization of all deflators under G is fully elaborated. The results are illustrated on the case of jump-diffusion model and the discrete-time market model.

Suggested Citation

  • Tahir Choulli & Ferdoos Alharbi, 2022. "Representation for martingales living after a random time with applications," Papers 2203.11072, arXiv.org, revised Nov 2022.
    • Handle: RePEc:arx:papers:2203.11072
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    References listed on IDEAS

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    1. 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.
    2. Anna Aksamit & Tahir Choulli & Jun Deng & Monique Jeanblanc, 2018. "No-arbitrage under a class of honest times," Finance and Stochastics, Springer, vol. 22(1), pages 127-159, January.
    3. Ferdoos Alharbi & Tahir Choulli, 2022. "Log-optimal portfolio after a random time: Existence, description and sensitivity analysis," Papers 2204.03798, arXiv.org.
    4. Tahir Choulli & Sina Yansori, 2018. "Explicit description of all deflators for market models under random horizon with applications to NFLVR," Papers 1803.10128, arXiv.org, revised Feb 2021.
    5. Claudio Fontana & Monique Jeanblanc & Shiqi Song, 2014. "On arbitrages arising with honest times," Finance and Stochastics, Springer, vol. 18(3), pages 515-543, July.
    6. Tahir Choulli & Sina Yansori, 2022. "Log-optimal and numéraire portfolios for market models stopped at a random time," Finance and Stochastics, Springer, vol. 26(3), pages 535-585, July.
    7. Alain BÉlanger & Steven E. Shreve & Dennis Wong, 2004. "A General Framework For Pricing Credit Risk," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 317-350, July.
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    Cited by:

    1. Ferdoos Alharbi & Tahir Choulli, 2022. "Log-optimal portfolio after a random time: Existence, description and sensitivity analysis," Papers 2204.03798, arXiv.org.

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