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Weak reflection principle for L\'evy processes

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  • Erhan Bayraktar
  • Sergey Nadtochiy

Abstract

In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is obtained by weakening the assumptions of symmetry required for the classical reflection principle to work. We call this method a weak reflection principle and show that it provides solutions to many problems for which the classical reflection principle is typically used. In addition, unlike the classical reflection principle, the new method works for a much larger class of stochastic processes which, in particular, do not possess any strong symmetries. Here, we review the existing results which establish the weak reflection principle for a large class of time-homogeneous diffusions on a real line and then proceed to extend this method to the L\'{e}vy processes with one-sided jumps (subject to some admissibility conditions). Finally, we demonstrate the applications of the weak reflection principle in financial mathematics, computational methods and inverse problems.

Suggested Citation

  • Erhan Bayraktar & Sergey Nadtochiy, 2013. "Weak reflection principle for L\'evy processes," Papers 1308.2250, arXiv.org, revised Oct 2015.
    • Handle: RePEc:arx:papers:1308.2250
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    References listed on IDEAS

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    1. Tehranchi, Michael R., 2009. "Symmetric martingales and symmetric smiles," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3785-3797, October.
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    Cited by:

    1. Sergey Nadtochiy & Jan Obloj, 2016. "Robust Trading of Implied Skew," Papers 1611.05518, arXiv.org.
    2. Jiro Akahori & Flavia Barsotti & Yuri Imamura, 2018. "Asymptotic Static Hedge via Symmetrization," Papers 1801.04045, arXiv.org.
    3. Sergey Nadtochiy & Jan Obłój, 2017. "Robust Trading Of Implied Skew," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(02), pages 1-41, March.

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