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A generalised Itō formula for Lévy-driven Volterra processes

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  • Bender, Christian
  • Knobloch, Robert
  • Oberacker, Philip

Abstract

We derive a generalised Itō formula for stochastic processes which are constructed by a convolution of a deterministic kernel with a centred Lévy process. This formula has a unifying character in the sense that it contains the classical Itō formula for Lévy processes as well as recent change-of-variable formulas for Gaussian processes such as fractional Brownian motion as special cases. Our result also covers fractional Lévy processes (with Mandelbrot–Van Ness kernel) and a wide class of related processes for which such a generalised Itō formula has not yet been available in the literature.

Suggested Citation

  • Bender, Christian & Knobloch, Robert & Oberacker, Philip, 2015. "A generalised Itō formula for Lévy-driven Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 2989-3022.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:8:p:2989-3022
    DOI: 10.1016/j.spa.2015.02.009
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    References listed on IDEAS

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    1. Barndorff-Nielsen, Ole E. & Benth, Fred Espen & Pedersen, Jan & Veraart, Almut E.D., 2014. "On stochastic integration for volatility modulated Lévy-driven Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 812-847.
    2. Christian Bender & Alexander Lindner & Markus Schicks, 2012. "Finite Variation of Fractional Lévy Processes," Journal of Theoretical Probability, Springer, vol. 25(2), pages 594-612, June.
    3. Neuman, Eyal, 2014. "The multifractal nature of Volterra–Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 3121-3145.
    4. Bender, Christian, 2014. "Backward SDEs driven by Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2892-2916.
    5. Basse, Andreas & Pedersen, Jan, 2009. "Lévy driven moving averages and semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2970-2991, September.
    6. Bender, Christian, 2003. "An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter," Stochastic Processes and their Applications, Elsevier, vol. 104(1), pages 81-106, March.
    7. Ole E. Barndorff-Nielsen & Fred Espen Benth & Almut E. D. Veraart, 2013. "Modelling energy spot prices by volatility modulated L\'{e}vy-driven Volterra processes," Papers 1307.6332, arXiv.org.
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    Cited by:

    1. Pakkanen, Mikko S. & Sottinen, Tommi & Yazigi, Adil, 2017. "On the conditional small ball property of multivariate Lévy-driven moving average processes," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 749-782.

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