# A fractional counting process and its connection with the Poisson process

01 Jan 2016-ALEA-Latin American Journal of Probability and Mathematical Statistics (Institute for Applied and Pure Mathematics (IMPA))-Vol. 13, Iss: 1, pp 291-307

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TL;DR: Both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractionsal compound Poissonprocess are given.

Abstract: 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.

17 citations

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TL;DR: This paper compares the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases; it is found that the FDE only outperformed the ODE in one of out three data sets.

Abstract: 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.

10 citations

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TL;DR: In this paper, the authors study non-homogeneous versions of the space-fractional and the time-frractional Poisson processes, which can be defined by means of the so-called multistable subordinator (a jump process with non-stationary increments), denoted by H. The authors consider the Poisson process time-changed by H and obtain its explicit distribution and governing equation.

Abstract: The space-fractional and the time-fractional Poisson processes are two well-known models of fractional evolution. They can be constructed as standard Poisson processes with the time variable replaced by a stable subordinator and its inverse, respectively. The aim of this paper is to study non-homogeneous versions of such models, which can be defined by means of the so-called multistable subordinator (a jump process with non-stationary increments), denoted by H. Firstly, we consider the Poisson process time-changed by H and we obtain its explicit distribution and governing equation. Then, by using the right-continuous inverse of H, we define an inhomogeneous analogue of the time-fractional Poisson process.

9 citations

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TL;DR: In this paper, strong solutions for two difference-differential fractional equations, defined via the generator of an immigration-death process, using spectral methods, were studied and the limit distribution of the time-changed process was analyzed.

Abstract: In this paper we study explicit strong solutions for two difference-differential fractional equations, defined via the generator of an immigration-death process, by using spectral methods. Moreover, we give a stochastic representation of the solutions of such difference-differential equations by means of a stable time-changed immigration-death process and we use this stochastic representation to show boundedness and then uniqueness of these strong solutions. Finally, we study the limit distribution of the time-changed process.

8 citations

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TL;DR: This work analyses 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.

Abstract: 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.

8 citations

##### References

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TL;DR: In this article, the authors introduced fractional derivatives of order α in time, with 0 for relaxation, diffusion, oscillations, and wave propagation, and showed that they are governed by simple differential equations of order 1 and 2 in time.

Abstract: The processes involving the basic phenomena of relaxation, diffusion, oscillations and wave propagation are of great relevance in physics; from a mathematical point of view they are known to be governed by simple differential equations of order 1 and 2 in time. The introduction of fractional derivatives of order α in time, with 0

842 citations

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01 Jan 1971

TL;DR: In this article, a linear operator of order functions of order (1.2) is defined and an operator of fractional integration is employed to prove results on the solutions of the integral equation.

Abstract: is an entire function of order $({\rm Re}\alpha)^{-1}$ and contains several well-known special functions as particular cases. We define a linear operator $\mathfrak{C}(\alpha, \beta;\rho;\lambda)$ on a space $L$ of functions by the integral in (1.2) and employ an operator of fractional integration $I^{\mu}$ : $L\rightarrow L$ to prove results on $\mathfrak{C}(\alpha, \beta;\rho;\lambda)$ ; these results are subsequently used to discuss theorems on the solutions of (1.2). The technique used can be apPlied to obtain analogous results on the integral equation

689 citations

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TL;DR: In this paper, it was shown that the fundamental solutions of the basic Cauchy and signalling problems can be expressed in terms of an auxiliary function M(z;β), where z = |x| t β is the similarity variable.

Abstract: The time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order 2β with 0 1 2 or 1 2 , respectively. Using the method of the Laplace transform, it is shown that the fundamental solutions of the basic Cauchy and Signalling problems can be expressed in terms of an auxiliary function M(z;β), where z = |x| t β is the similarity variable. Such function is proved to be an entire function of Wright type.

573 citations

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19 Mar 2008

TL;DR: In this article, Mittag-Leffler functions and fractional calculus are used for estimating density and order statistics in time series and wavelet analysis, respectively, in the context of matrix arguments.

Abstract: Basic Ideas of Special Functions and Statistical Distributions.- Mittag-Leffler Functions and Fractional Calculus.- An Introduction to q-Series.- Ramanujan's Theories of Theta and Elliptic Functions.- Lie Group and Special Functions.- Applications to Stochastic Process and Time Series.- Applications to Density Estimation.- Applications to Order Statistics.- Applications to Astrophysics Problems.- An Introduction to Wavelet Analysis.- Jacobians of Matrix Transformations.- Special Functions of Matrix Argument.

388 citations

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TL;DR: In this article, a fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov-Feller equation.

Abstract: A fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov–Feller equation. We have found the probability of n arrivals by time t for fractional stream of events. The fractional Poisson process captures long-memory effect which results in non-exponential waiting time distribution empirically observed in complex systems. In comparison with the standard Poisson process the developed model includes additional parameter μ. At μ=1 the fractional Poisson becomes the standard Poisson and we reproduce the well known results related to the standard Poisson process. As an application of developed fractional stochastic model we have introduced and elaborated fractional compound Poisson process.

269 citations