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Fractional smoothness and applications in Finance

Author

Listed:
  • Stefan Geiss

    (Department of Mathematics [Innsbruck] - Leopold Franzens Universität Innsbruck - University of Innsbruck)

  • Emmanuel Gobet

    (MATHFI - Mathématiques financières - LJK - Laboratoire Jean Kuntzmann - UPMF - Université Pierre Mendès France - Grenoble 2 - UJF - Université Joseph Fourier - Grenoble 1 - Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology - CNRS - Centre National de la Recherche Scientifique)

Abstract

This overview article concerns the notion of fractional smoothness of random variables of the form $g(X_T)$, where $X=(X_t)_{t\in [0,T]}$ is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance mainly concern the analysis of discrete time hedging errors. We close the review by indicating some further developments.

Suggested Citation

  • Stefan Geiss & Emmanuel Gobet, 2011. "Fractional smoothness and applications in Finance," Post-Print hal-00474803, HAL.
    • Handle: RePEc:hal:journl:hal-00474803
    DOI: 10.1007/978-3-642-18412-3_12
    Note: View the original document on HAL open archive server: https://hal.science/hal-00474803
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    References listed on IDEAS

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    1. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    2. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    3. Geiss, Christel & Geiss, Stefan, 2006. "On an approximation problem for stochastic integrals where random time nets do not help," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 407-422, March.
    4. Mats Brod'en & Peter Tankov, 2010. "Tracking errors from discrete hedging in exponential L\'evy models," Papers 1003.0709, arXiv.org.
    5. Takaki Hayashi & Per A. Mykland, 2005. "Evaluating Hedging Errors: An Asymptotic Approach," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 309-343, April.
    6. Emmanuel Temam & Emmanuel Gobet, 2001. "Discrete time hedging errors for options with irregular payoffs," Finance and Stochastics, Springer, vol. 5(3), pages 357-367.
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