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The Fractional Poisson Process and the Inverse Stable Subordinator
Ear,Nih grant R Eb +1 more
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TLDR
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.Abstract:
The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve
a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a
traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a
fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional
diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered
fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.read more
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Mixtures of Tempered Stable Subordinators
TL;DR: In this paper, the authors introduced mixtures of tempered stable subordinators (TSS) and defined a class of subordinators which generalize TSS, and generalized these results to n-th order mixtures.
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Fractional discrete processes: compound and mixed Poisson representations
Luisa Beghin,Claudio Macci +1 more
TL;DR: Two fractional versions of a family of nonnegative integer valued processes are considered, it is proved that their probability mass functions solve fractional Kolmogorov forward equations, and the overdispersion of these processes is shown.
Dissertation
Fractal activity time and integer valued models in finance
TL;DR: In this article, the authors propose a model to accurately describe the price evolution of a risky asset, a security traded on a financial market such as a stock, currency or benchmark index.
Book ChapterDOI
Continuous time random walks and space-time fractional differential equations
TL;DR: The continuous time random walk as mentioned in this paper is a model from statistical physics that elucidates the physical interpretation of the space-time fractional diffusion equation, where each step in the random walk is separated by a random waiting time.
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On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators
Khrystyna Buchak,Lyudmyla Sakhno +1 more
TL;DR: In this article, the governing equations for marginal distributions of Poisson and Skellam processes were presented in terms of convolution-type derivatives, and the equations were given by using inverse subordinators.
References
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Journal ArticleDOI
Correlation Structure of Time-Changed Lévy Processes
TL;DR: In this article, the correlation function for time-changed L evy processes has been studied in the context of continuous time random walks, where the second-order correlation function of a continuous-time random walk is defined.
Journal ArticleDOI
Applications of inverse tempered stable subordinators
TL;DR: This paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time-fractional diffusion equation, and its “folded” density solves a temperamental telegraph equation.
Journal ArticleDOI
Time-changed Poisson processes
TL;DR: In this article, the authors considered time-changed Poisson processes and derived the governing difference-differential equations (DDEs) for these processes, and derived a new governing partial differential equation for the tempered stable subordinator of index 0 β 1.
Journal ArticleDOI
Fractional Skellam processes with applications to finance
TL;DR: 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.
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Inverse Tempered Stable Subordinators
TL;DR: In this paper, the first-hitting time of a tempered β-stable subordinator, also called inverse tempered stable (ITS) subordinator is considered, and the limiting form of the ITS density, as the space variable $x\rightarrow 0$, and its $k$-th order derivatives are obtained.