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A Smooth Approach to Malliavin Calculus for Lévy Processes

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  • Horst Osswald

    (Mathematisches Institut der Universität München)

Abstract

An approach to Malliavin calculus for Lévy processes, discrete in time and smooth in chance, is presented. Each Lévy triple can be satisfied by a Lévy process living on a fixed sample space Ω, which is, in a certain sense, a finite dimensional Euclidean space. The probability measures on Ω characterize the Lévy processes. We compare these measures with the associated Lévy measures, and present several examples. Using chaos expansions for Lévy functionals, even for those having no moments, we can represent all these functionals by polynomials in several variables. There exists an effective method to compute the kernels of the chaos decomposition. Finally, we point out several applications, which are postponed to a succession of papers.

Suggested Citation

  • Horst Osswald, 2009. "A Smooth Approach to Malliavin Calculus for Lévy Processes," Journal of Theoretical Probability, Springer, vol. 22(2), pages 441-473, June.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:2:d:10.1007_s10959-008-0148-8
    DOI: 10.1007/s10959-008-0148-8
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    References listed on IDEAS

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    1. Solé, Josep Lluís & Utzet, Frederic & Vives, Josep, 2007. "Canonical Lévy process and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 165-187, February.
    2. Nualart, David & Schoutens, Wim, 2000. "Chaotic and predictable representations for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 90(1), pages 109-122, November.
    3. Josep Vives & Jorge A. León & Frederic Utzet & Josep L. Solé, 2002. "On Lévy processes, Malliavin calculus and market models with jumps," Finance and Stochastics, Springer, vol. 6(2), pages 197-225.
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    Cited by:

    1. Horst Osswald, 2009. "On Anticipative Girsanov Transformations for Lévy Processes," Journal of Theoretical Probability, Springer, vol. 22(2), pages 474-481, June.

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