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A superhedging approach to stochastic integration

Author

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  • Łochowski, Rafał M.
  • Perkowski, Nicolas
  • Prömel, David J.

Abstract

Using Vovk’s outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for “typical price paths” in the space of càdlàg functions possessing a mild restriction on the jumps directed downwards. In particular, this result includes the existence of quadratic variation of “typical price paths” in the space of non-negative càdlàg paths and implies the existence of quadratic variation in the sense of Föllmer quasi surely under all martingale measures. Based on the robust existence of the quadratic variation, a model-free Itô integration is developed.

Suggested Citation

  • Łochowski, Rafał M. & Perkowski, Nicolas & Prömel, David J., 2018. "A superhedging approach to stochastic integration," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4078-4103.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:12:p:4078-4103
    DOI: 10.1016/j.spa.2018.01.009
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    References listed on IDEAS

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    1. Alexander Schied & Iryna Voloshchenko, 2015. "Pathwise no-arbitrage in a class of Delta hedging strategies," Papers 1511.00026, arXiv.org, revised Jun 2016.
    2. Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.
    3. Willinger, Walter & Taqqu, Murad S., 1989. "Pathwise stochastic integration and applications to the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 253-280, August.
    4. Mark Davis & Jan Obłój & Vimal Raval, 2014. "Arbitrage Bounds For Prices Of Weighted Variance Swaps," Mathematical Finance, Wiley Blackwell, vol. 24(4), pages 821-854, October.
    5. Nicolas Perkowski & David J. Promel, 2014. "Local times for typical price paths and pathwise Tanaka formulas," Papers 1405.4421, arXiv.org, revised Apr 2015.
    6. Karandikar, Rajeeva L., 1995. "On pathwise stochastic integration," Stochastic Processes and their Applications, Elsevier, vol. 57(1), pages 11-18, May.
    7. Bick, Avi & Willinger, Walter, 1994. "Dynamic spanning without probabilities," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 349-374, April.
    8. Nicolas Perkowski & David J. Promel, 2013. "Pathwise stochastic integrals for model free finance," Papers 1311.6187, arXiv.org, revised Jun 2016.
    9. Mathias Beiglböck & Alexander M. G. Cox & Martin Huesmann & Nicolas Perkowski & David J. Prömel, 2017. "Pathwise superreplication via Vovk’s outer measure," Finance and Stochastics, Springer, vol. 21(4), pages 1141-1166, October.
    10. Zhaoxu Hou & Jan Obloj, 2015. "On robust pricing-hedging duality in continuous time," Papers 1503.02822, arXiv.org, revised Jul 2015.
    11. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
    12. Christa Cuchiero & Walter Schachermayer & Ting-Kam Leonard Wong, 2016. "Cover's universal portfolio, stochastic portfolio theory and the numeraire portfolio," Papers 1611.09631, arXiv.org.
    13. Candia Riga, 2016. "A pathwise approach to continuous-time trading," Papers 1602.04946, arXiv.org.
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    Citations

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    Cited by:

    1. Andrew L. Allan & Chong Liu & David J. Promel, 2021. "A C\`adl\`ag Rough Path Foundation for Robust Finance," Papers 2109.04225, arXiv.org, revised May 2023.
    2. Rafa{l} M. {L}ochowski & Nicolas Perkowski & David J. Promel, 2021. "One-dimensional game-theoretic differential equations," Papers 2101.08041, arXiv.org.
    3. Lesiba Ch. Galane & Rafa{l} M. {L}ochowski & Farai J. Mhlanga, 2018. "On SDEs with Lipschitz coefficients, driven by continuous, model-free martingales," Papers 1807.05692, arXiv.org, revised Feb 2022.
    4. Rafa{l} M. {L}ochowski, 2021. "BDG inequalities and their applications for model-free continuous price paths with instant enforcement," Papers 2109.07928, arXiv.org, revised Aug 2023.

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