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Tempered Stable Distributions

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

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Abstract

In this chapter we discuss two applications of tempered stable distributions. The first is to option pricing and the second is to mobility models. We also discuss the mechanism by which tempered stable distributions appear in applications.

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Notes

  1. 1.

    This means that for any \(A \in \mathcal{F}\) we have P(A) = 0 if and only if Q(A) = 0.

  2. 2.

    Although the Radon-Nikodym derivative process only defines measures on \((\varOmega,\mathcal{F}_{t})\) for t ≥ 0, it, in fact, uniquely determines a probability measure on \((\varOmega,\mathcal{F})\). See the discussion near Definition 33.4 in [69].

  3. 3.

    This follows from properties of Poisson random measures and the fact that the jump measure of a Lévy process is a Poisson random measure, see, e.g., Theorem 19.2 in [69] or Chapter 3 in [21].

  4. 4.

    A version of this result also appears in [33]. It should be noted that we are using a slightly different parametrization than the one used in [33] and [34].

  5. 5.

    It should be noted that [34] reported a slightly different natural scale. This is due to that fact that a different parametrization was used there.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of North Carolina, Charlotte, NC, USA

    Michael Grabchak

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  1. Michael Grabchak
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© 2016 Michael Grabchak

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Grabchak, M. (2016). Applications. In: Tempered Stable Distributions. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-24927-8_7

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