A fractional counting process and its connection with the
Poisson process

The fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

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“…Proof. According to Proposition 4.1 from Di Crescenzo et al (2016) we have the result that for fixed t > 0 the convergence…”

Section: A One-dimensional Limit Resultsmentioning

confidence: 96%

Limit theorems for the fractional nonhomogeneous Poisson process

2019

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“…3 (t)) which is simply the case when α = 1. Interestingly, the fractional stochastic SEIR process is a random-time subordination of the integer stochastic SEIR model, as established for other fractional processes like the fractional Poisson process [37,45,52], and the fractional birth and/or death processes [39,40,42,43,53]. In Mandelbrot and Taylor [54], the stochastic time process T 2α is called the operational time, and t is the physical time.…”

Section: Transitionmentioning

confidence: 99%

“…with p (α) (i,j,k) (t) = 0 if either i, j, or k are negative or i + j + k > N (see Di Crescenzo et al [45]). The classical forward Kolmogorov equation of the stochastic SEIR model follows when α = 1 with state probabilities p (1) (i,j,k) (t), [51] (p. 321).…”

Section: Transitionmentioning

confidence: 99%

“…Saeedian et al [36] showed how another memory functional of the process can lead to replacing the integer derivatives with Caputo fractional derivatives. In this paper, we show how Caputo fractional differential equations follow naturally from fractional stochastic processes like those introduced in literature [37][38][39][40][41][42][43][44][45][46]. We then compare transient and long term dynamics between the FDE and ODE models while fitting them to three different data sets.…”

Section: Introductionmentioning

confidence: 99%

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Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

Islam

Peace

Medina

et al. 2020

IJERPH

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In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations (FDEs) as approximations of some type of fractional nonlinear birth and death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While ordinary differential equations (ODEs) are commonly used to model epidemics, FDEs are more flexible in fitting empirical data and theoretically offer improved model predictions. The question arises whether, in practice, the benefits of using FDEs over ODEs outweigh the added computational complexities. While important differences in transient dynamics were observed, the FDE only outperformed the ODE in one of out three data sets. In general, FDE modeling approaches may be worth it in situations with large refined data sets and good numerical algorithms. Laplace transform of a function f (t) is defined as L( f )(s) =f (s) = ∞ 0 e −st f (t)dt.

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“…In the recent past, pure-jump fractional processes have attracted great attention. Just to mention a few examples, Beghin and Orsingher [3] illustrated various results on the fractional Poisson process and also focused on certain higher-order extensions, whilst a fractional counting process with multiple jumps was studied in [11]. See also [33] for a generalization of the space-fractional Poisson process.…”

Section: Introductionmentioning

confidence: 99%

On a jump-telegraph process driven by an alternating fractional Poisson process

Crescenzo

Meoli

2018

J. Appl. Probab.

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The basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds.

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A fractional counting process and its connection with the Poisson process

Cited by 29 publications

References 34 publications

Limit theorems for the fractional nonhomogeneous Poisson process

Limit theorems for the fractional nonhomogeneous Poisson process

Integer Versus Fractional Order SEIR Deterministic and Stochastic Models of Measles

On a jump-telegraph process driven by an alternating fractional Poisson process

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