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Functionals of a Lévy Process on Canonical and Generic Probability Spaces

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  • Alexander Steinicke

    (University of Innsbruck)

Abstract

We develop an approach to Malliavin calculus for Lévy processes from the perspective of expressing a random variable $$Y$$ Y by a functional $$F$$ F mapping from the Skorohod space of càdlàg functions to $$\mathbb {R}$$ R , such that $$Y=F(X)$$ Y = F ( X ) where $$X$$ X denotes the Lévy process. We also present a chain-rule-type application for random variables of the form $$f(\omega ,Y(\omega ))$$ f ( ω , Y ( ω ) ) . An important tool for these results is a technique which allows us to transfer identities proved on the canonical probability space (in the sense of Solé et al.) associated to a Lévy process with triplet $$(\gamma ,\sigma ,\nu )$$ ( γ , σ , ν ) to an arbitrary probability space $$(\varOmega ,\mathcal {F},\mathbb {P})$$ ( Ω , F , P ) which carries a Lévy process with the same triplet.

Suggested Citation

  • Alexander Steinicke, 2016. "Functionals of a Lévy Process on Canonical and Generic Probability Spaces," Journal of Theoretical Probability, Springer, vol. 29(2), pages 443-458, June.
    • Handle: RePEc:spr:jotpro:v:29:y:2016:i:2:d:10.1007_s10959-014-0583-7
    DOI: 10.1007/s10959-014-0583-7
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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. Delong, Lukasz & Imkeller, Peter, 2010. "On Malliavin's differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1748-1775, August.
    3. Ishikawa, Yasushi & Kunita, Hiroshi, 2006. "Malliavin calculus on the Wiener-Poisson space and its application to canonical SDE with jumps," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1743-1769, December.
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    Cited by:

    1. Stefan Kremsner & Alexander Steinicke, 2022. "$${{\varvec{L}}}^{{\varvec{p}}}$$ L p -Solutions and Comparison Results for Lévy-Driven Backward Stochastic Differential Equations in a Monotonic, General Growth Setting," Journal of Theoretical Probability, Springer, vol. 35(1), pages 231-281, March.

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