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Fractional Skellam processes with applications to finance
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In this paper, the authors define fractional Skellam processes via the time changes in Poisson and Skekam processes by an inverse of a standard stable subordinator.Abstract:
The recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Leffler distribution of inter-arrival times. This paper defines fractional Skellam processes via the time changes in Poisson and Skellam processes by an inverse of a standard stable subordinator. An application to high frequency financial data set is provided to illustrate the advantages of models based on fractional Skellam processes.read more
Citations
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Special Functions of Mathematical Physics and Chemistry. By Ian N. Sneddon. Pp. 164. 10s. 6d. 1956. (Oliver and Boyd)
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On History of Mathematical Economics: Application of Fractional Calculus
TL;DR: A review of the history of applications of fractional calculus in modern mathematical economics and economic theory can be found in this article, where the main mathematical tool designed to "cure amnesia" in economics is a theory of integrals, derivatives, sums, and differences of non-integer orders.
Journal ArticleDOI
Concept of dynamic memory in economics
TL;DR: Some general restrictions that can be imposed on the structure and properties of dynamic memory are described and a generalization of the Harrod–Domar model, where the power-law memory is taken into account, is considered.
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Modeling of financial processes with a space-time fractional diffusion equation of varying order
TL;DR: In this paper, a new model for financial processes in form of a space-time fractional diffusion equation of varying order is introduced, analyzed, and applied for some financial data.
The Fractional Poisson Process and the Inverse Stable Subordinator
Ear,Nih grant R Eb +1 more
TL;DR: In this paper, it was shown that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional poisson process with Mittag-Leffler waiting times.
References
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Journal ArticleDOI
Mittag-Leffler Functions and Their Applications
TL;DR: In this survey paper, nearly all types of Mittag-Leffler type functions existing in the literature are presented and an attempt is made to present nearly an exhaustive list of references to make the reader familiar with the present trend of research in Mittag, Leffler, and type functions and their applications.
Book
Stochastic Models for Fractional Calculus
TL;DR: In this article, the traditional diffusion model was extended to the vector fractional diffusion model, which is the state-of-the-art diffusion model for the problem of diffusion.
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On Mittag-Leffler-type functions in fractional evolution processes
TL;DR: In this article, a variety of fractional evolution processes are reviewed, whose solutions turn out to be related to Mittag-Leffler-type functions, and the chosen equations are the simplest of the fractional calculus and include the Abel integral equations of the second kind.
Posted Content
Mittag-Leffler Functions and Their Applications
TL;DR: A detailed survey of Mittag-Leffler type functions can be found in this article, where the authors present a detailed account or rather a brief survey of the Mittag Leffler function, generalized Mittag leffler functions and their interesting and useful properties.
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Modelling microstructure noise with mutually exciting point processes
TL;DR: In this paper, the authors introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 and 2 for a single asset and a pair of assets.