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Random variables as pathwise integrals with respect to fractional Brownian motion

Author

Listed:
  • Mishura, Yuliya
  • Shevchenko, Georgiy
  • Valkeila, Esko

Abstract

We give both necessary and sufficient conditions for a random variable to be represented as a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand. We also show that any random variable is a value of such integral in an improper sense and that such integral can have any prescribed distribution. We discuss some applications of these results, in particular, to fractional Black–Scholes model of financial market.

Suggested Citation

  • Mishura, Yuliya & Shevchenko, Georgiy & Valkeila, Esko, 2013. "Random variables as pathwise integrals with respect to fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2353-2369.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:6:p:2353-2369
    DOI: 10.1016/j.spa.2013.02.015
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    References listed on IDEAS

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    1. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    2. Christian Bender & Tommi Sottinen & Esko Valkeila, 2010. "Fractional processes as models in stochastic finance," Papers 1004.3106, arXiv.org.
    3. Azmoodeh Ehsan & Mishura Yuliya & Valkeila Esko, 2009. "On hedging European options in geometric fractional Brownian motion market model," Statistics & Risk Modeling, De Gruyter, vol. 27(02), pages 129-144, December.
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    Cited by:

    1. Mishura, Yuliya & Shevchenko, Georgiy, 2017. "Small ball properties and representation results," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 20-36.
    2. Yaskov, Pavel, 2018. "Extensions of the sewing lemma with applications," Stochastic Processes and their Applications, Elsevier, vol. 128(11), pages 3940-3965.

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