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Canonical Lévy process and Malliavin calculus

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  • Solé, Josep Lluís
  • Utzet, Frederic
  • Vives, Josep

Abstract

A suitable canonical Lévy process is constructed in order to study a Malliavin calculus based on a chaotic representation property of Lévy processes proved by Itô using multiple two-parameter integrals. In this setup, the two-parameter derivative Dt,x is studied, depending on whether x=0 or x[not equal to]0; in the first case, we prove a chain rule; in the second case, a formula by trajectories.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:117:y:2007:i:2:p:165-187
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    References listed on IDEAS

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    1. 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.
    2. Davis, Mark H.A. & Johansson, Martin P., 2006. "Malliavin Monte Carlo Greeks for jump diffusions," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 101-129, January.
    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:

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    2. Ankirchner, Stefan, 2008. "On filtration enlargements and purely discontinuous martingales," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1662-1678, September.
    3. Laukkarinen, Eija, 2020. "Malliavin smoothness on the Lévy space with Hölder continuous or BV functionals," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4766-4792.
    4. Masahiro Handa & Noriyoshi Sakuma & Ryoichi Suzuki, 2024. "A Girsanov transformed Clark-Ocone-Haussmann type formula for $$L^1$$ L 1 -pure jump additive processes and its application to portfolio optimization," Annals of Finance, Springer, vol. 20(3), pages 329-352, September.
    5. Takuji Arai & Yuto Imai, 2017. "A closed-form representation of mean-variance hedging for additive processes via Malliavin calculus," Papers 1702.07556, arXiv.org, revised Nov 2017.
    6. Suzuki, Ryoichi, 2018. "Malliavin differentiability of indicator functions on canonical Lévy spaces," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 183-190.
    7. El-Khatib, Youssef & Goutte, Stephane & Makumbe, Zororo S. & Vives, Josep, 2023. "A hybrid stochastic volatility model in a Lévy market," International Review of Economics & Finance, Elsevier, vol. 85(C), pages 220-235.
    8. 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.
    9. Takuji Arai & Ryoichi Suzuki, 2019. "A Clark-Ocone type formula via Ito calculus and its application to finance," Papers 1906.06648, arXiv.org.
    10. Eden, Richard & Víquez, Juan, 2015. "Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 182-216.
    11. 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.
    12. Jin, Sixian & Schellhorn, Henry & Vives, Josep, 2020. "Dyson type formula for pure jump Lévy processes with some applications to finance," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 824-844.
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    15. Murr, Rüdiger, 2013. "Characterization of infinite divisibility by duality formulas. Application to Lévy processes and random measures," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1729-1749.
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    18. Choe, Hi Jun & Lee, Ji Min & Lee, Jung-Kyung, 2018. "Malliavin calculus for subordinated Lévy process," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 392-401.

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