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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:

    1. Fujii, Masaaki & Takahashi, Akihiko, 2018. "Quadratic–exponential growth BSDEs with jumps and their Malliavin’s differentiability," Stochastic Processes and their Applications, Elsevier, vol. 128(6), pages 2083-2130.
    2. Takuji Arai & Yuto Imai & Ryoichi Suzuki, 2017. "Local risk-minimization for Barndorff-Nielsen and Shephard models," Finance and Stochastics, Springer, vol. 21(2), pages 551-592, April.
    3. Takuji Arai & Yuto Imai & Ryo Nakashima, 2018. "Numerical analysis on quadratic hedging strategies for normal inverse Gaussian models," Papers 1801.05597, arXiv.org.
    4. Ankirchner, Stefan, 2008. "On filtration enlargements and purely discontinuous martingales," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1662-1678, September.
    5. 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.
    6. Horst Osswald, 2009. "A Smooth Approach to Malliavin Calculus for Lévy Processes," Journal of Theoretical Probability, Springer, vol. 22(2), pages 441-473, June.
    7. 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.
    8. 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.
    9. Suzuki, Ryoichi, 2018. "Malliavin differentiability of indicator functions on canonical Lévy spaces," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 183-190.
    10. 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.
    11. 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.
    12. Takuji Arai & Ryoichi Suzuki, 2019. "A Clark-Ocone type formula via Ito calculus and its application to finance," Papers 1906.06648, arXiv.org.
    13. Bernardo D'Auria & Jos'e A. Salmer'on, 2021. "Anticipative information in a Brownian-Poissonmarket: the binary information," Papers 2111.01529, arXiv.org.
    14. 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.
    15. 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.
    16. 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.
    17. 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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