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Gamma Kernels and BSS/LSS Processes

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Advanced Modelling in Mathematical Finance

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 189))

Abstract

This paper reviews the roles of gamma type kernels in the theory and modelling for Brownian and Lévy semistationary processes. Applications to financial econometrics and the physics of turbulence are pointed out.

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Notes

  1. 1.

    Proofs and technical details are, in most cases, not presented here. For these and related literature we refer to the papers cited.

  2. 2.

    The differentiation term \(DD^{n,\gamma }\) may be viewed as a special case of the more general definition

    $$\begin{aligned} D^{m,n,\gamma }=D^{m}D^{n,\gamma } \end{aligned}$$

    where m, like n, is a nonnegative integer and \(0<\gamma <1\). Then \( D^{m,0,\gamma }\) equals the Riemann-Liouville fractional derivative while \( D^{0,n,\gamma }\) is the Caputo fractional derivative.

  3. 3.

    The \(-5/3\) behaviour is very manifest in the socalled inertial range but, as documented by later, extensive and accurate measurements, for the largest frequencies (the dissipation range) the spectral density decreases at a much faster rate. The total behaviour of the spectral density is accurately described by a formula due to Skharofsky, see for instance Fig. 5 in [35].

  4. 4.

    The Hypothesis states that spatial and temporal increments of the main component of the velocity vector are equivalent in law up to a proportional change of time. Cf. for instance [43].

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Acknowledgements

I am grateful to the reviewers of the paper and to the editors for their helpful comments and suggestions regarding the original version of my manuscript and to Orimar Sauri for a careful checking of the revised version.

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Authors and Affiliations

  1. Department of Mathematics and CREATES (Funded by the Danish National Research Foundation Under Grant DNRF78), Thiele Centre, University of Aarhus, Aarhus, Denmark

    Ole E. Barndorff-Nielsen

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  1. Ole E. Barndorff-Nielsen
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Correspondence to Ole E. Barndorff-Nielsen .

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Editors and Affiliations

  1. Department of Mathematics, Christian-Albrechts-Universität zu Kiel, Kiel, Germany

    Jan Kallsen

  2. Institute of Mathematics, Technische Universität Berlin, Berlin, Germany

    Antonis Papapantoleon

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Barndorff-Nielsen, O.E. (2016). Gamma Kernels and BSS/LSS Processes. In: Kallsen, J., Papapantoleon, A. (eds) Advanced Modelling in Mathematical Finance. Springer Proceedings in Mathematics & Statistics, vol 189. Springer, Cham. https://doi.org/10.1007/978-3-319-45875-5_2

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