Fractional Poisson Process Research Articles

Fractional Poisson Process: Long-Range Dependence and Applications in Ruin Theory

We study a renewal risk model in which the surplus process of the insurance company is modelled by a compound fractional Poisson process. We establish the long-range dependence property of this nonstationary process. Some results for ruin probabilities are presented under various assumptions on the distribution of the claim sizes.

Fractional Poisson Process: Long-Range Dependence and Applications in Ruin Theory

We study a renewal risk model in which the surplus process of the insurance company is modelled by a compound fractional Poisson process. We establish the long-range dependence property of this nonstationary process. Some results for ruin probabilities are presented under various assumptions on the distribution of the claim sizes.

Fractional Skellam processes with applications to finance

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.

Estimation of parameters in the fractional compound Poisson process

This paper discusses method-of-moments estimators for parameters in the fractional compound Poisson process and establishes asymptotic normality of estimators. Simulation are presented to illustrate the properties of the estimators.

A family of random walks with generalized Dirichlet steps

We analyze a class of continuous time random walks in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^d,d\ge 2,$\end{document}Rd,d≥2, with uniformly distributed directions. The steps performed by these processes are distributed according to a generalized Dirichlet law. Given the number of changes of orientation, we provide the analytic form of the probability density function of the position \documentclass[12pt]{minimal}\begin{document}$\lbrace \underline{\bf X}_d(t),t>0\rbrace$\end{document}{X̲d(t),t>0} reached, at time t > 0, by the random motion. In particular, we analyze the case of random walks with two steps. In general, it is a hard task to obtain the explicit probability distributions for the process \documentclass[12pt]{minimal}\begin{document}$\lbrace \underline{\bf X}_d(t),t>0\rbrace$\end{document}{X̲d(t),t>0}. Nevertheless, for suitable values for the basic parameters of the generalized Dirichlet probability distribution, we are able to derive the explicit conditional density functions of \documentclass[12pt]{minimal}\begin{document}$\lbrace \underline{\bf X}_d(t),t>0\rbrace$\end{document}{X̲d(t),t>0}. Furthermore, in some cases, by exploiting the fractional Poisson process, the unconditional probability distributions of the random walk are obtained. This paper extends in a more general setting, the random walks with Dirichlet displacements introduced in some previous papers.

Fractional Poisson process with random drift

We study the connection between PDEs and Levy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenius-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in some cases, the explicit representation of the transition probability laws. To this aim, we show that a special role is played by the translation operator associated to the representation of the Poisson semigroup.

The Fractional Differential Polynomial Neural Network for Approximation of Functions

In this work, we introduce a generalization of the differential polynomial neural network utilizing fractional calculus. Fractional calculus is taken in the sense of the Caputo differential operator. It approximates a multi-parametric function with particular polynomials characterizing its functional output as a generalization of input patterns. This method can be employed on data to describe modelling of complex systems. Furthermore, the total information is calculated by using the fractional Poisson process.

Fractional Poisson Fields

Using inverse subordinators and Mittag-Leffler functions, we present a new definition of a fractional Poisson process parametrized by points of the Euclidean space $\mathbb{R}_+^2$ . Some properties are given and, in particular, we prove a long-range dependence property.

On the integral of fractional Poisson processes

In this paper, we consider the Riemann–Liouville fractional integral Nα,ν(t)=1Γ(α)∫0t(t−s)α−1Nν(s)ds, where Nν(t), t≥0, is a fractional Poisson process of order ν∈(0,1], and α>0. We give the explicit bivariate distribution Pr{Nν(s)=k,Nν(t)=r}, for t≥s, r≥k, the mean ENα,ν(t) and the variance VarNα,ν(t). We study the process Nα,1(t) for which we are able to produce explicit results for the conditional and absolute variances and means. Much more involved results on N1,1(t) are presented in the last section where also distributional properties of the integrated Poisson process (including the representation as random sums) is derived. The integral of powers of the Poisson process is examined and its connections with generalized harmonic numbers are discussed.

Renewal processes based on generalized Mittag–Leffler waiting times

The fractional Poisson process has recently attracted experts from several fields of study. Its natural generalization of the ordinary Poisson process made the model more appealing for real-world applications. In this paper, we generalized the standard and fractional Poisson processes through the waiting time distribution, and showed their relations to an integral operator with a generalized Mittag–Leffler function in the kernel. The waiting times of the proposed renewal processes have the generalized Mittag–Leffler and stretched–squashed Mittag–Leffler distributions. Note that the generalizations naturally provide greater flexibility in modeling real-life renewal processes. Algorithms to simulate sample paths and to estimate the model parameters are derived. Note also that these procedures are necessary to make these models more usable in practice. State probabilities and other qualitative or quantitative features of the models are also discussed.

Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes

In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes $N_\alpha(t)$, $N_\beta(t)$, $t>0$, we show that $N_\alpha(N_\beta(t)) \overset{\text{d}}{=} \sum_{j=1}^{N_\beta(t)} X_j$, where the $X_j$s are Poisson random variables. We present a series of similar cases, the most general of which is the one in which the outer process is Poisson and the inner one is a nonlinear fractional birth process. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns compositions of the form $N_\alpha(\tau_k^\nu)$, $\nu \in (0,1]$, where $\tau_k^\nu$ is the inverse of the fractional Poisson process, and we show how these compositions can be represented as random sums. Furthermore we study compositions of the form $\Theta(N(t))$, $t>0$, which can be represented as random products. The last section is devoted to studying continued fractions of Cauchy random variables with a Poisson number of levels. We evaluate the exact distribution and derive the scale parameter in terms of ratios of Fibonacci numbers.

Analytic solutions of fractional differential equations by operational methods

We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation of differential equations of mathematical physics. Fractionality is obtained by substituting the ordinary integer-order derivative with the Caputo fractional derivative. Furthermore, operational relations between ordinary and fractional differentiation are shown and discussed in detail. Finally, a last example concerns the application of the method to the study of a fractional Poisson process.

Flying randomly in [formula omitted] with Dirichlet displacements

Random flights in R d , d ≥ 2 , with Dirichlet-distributed displacements and uniformly distributed orientation are analyzed. The explicit characteristic functions of the position X ¯ d ( t ) , t > 0 , when the number of changes of direction is fixed are obtained. The probability distributions are derived by inverting the characteristic functions for all dimensions d of R d and many properties of the probabilistic structure of X ¯ d ( t ) , t > 0 are examined. If the number of changes of direction is randomized by means of a fractional Poisson process, we are able to obtain explicit distributions for P { X ¯ d ( t ) ∈ d x ¯ d } for all d ≥ 2 . A section is devoted to random flights in R 3 where the general results are discussed. The existing literature is compared with the results of this paper where in our view classical Pearson’s problem of random flights is resolved by suitably randomizing the step lengths. The random flights where changes of direction are governed by a homogeneous Poisson process are analyzed and compared with the model of Dirichlet-distributed displacements of this work.

Non-exponential extinction of radiation by fractional calculus modelling

Possible deviations from exponential attenuation of radiation in a random medium have been recently studied in several works. These deviations from the classical Beer–Lambert law were justified from a stochastic point of view by Kostinski (2001) [1]. In his model he introduced the spatial correlation among the random variables, i.e. a space memory. In this note we introduce a different approach, including a memory formalism in the classical Beer–Lambert law through fractional calculus modelling. We find a generalized Beer–Lambert law in which the exponential memoryless extinction is only a special case of non-exponential extinction solutions described by Mittag–Leffler functions. We also justify this result from a stochastic point of view, using the space fractional Poisson process. Moreover, we discuss some concrete advantages of this approach from an experimental point of view, giving an estimate of the deviation from exponential extinction law, varying the optical depth. This is also an interesting model to understand the meaning of fractional derivative as an instrument to transmit randomness of microscopic dynamics to the macroscopic scale.

The Fractional Poisson Process and the Inverse Stable Subordinator

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.

Parameter estimation for fractional Poisson processes

The paper proposes a formal estimation procedure for parameters of the fractional Poisson process (fPp). Such procedures are needed to make the fPp model usable in applied situations. The basic idea of fPp, motivated by experimental data with long memory is to make the standard Poisson model more flexible by permitting non-exponential, heavy-tailed distributions of interarrival times and different scaling properties. We establish the asymptotic normality of our estimators for the two parameters appearing in our fPp model. This fact permits construction of the corresponding confidence intervals. The properties of the estimators are then tested using simulated data.

Fractional Poisson processes and related planar random motions

We present three different fractional versions of the Poisson process and some related results concerning the distribution of order statistics and the compound Poisson process. The main version is constructed by considering the difference-differential equation governing the distribution of the standard Poisson process, $ N(t),t>0$, and by replacing the time-derivative with the fractional Dzerbayshan-Caputo derivative of order $\nu\in(0,1]$. For this process, denoted by $\mathcal{N}_\nu(t),t>0,$ we obtain an interesting probabilistic representation in terms of a composition of the standard Poisson process with a random time, of the form $\mathcal{N}_\nu(t)= N(\mathcal{T}_{2\nu}(t)),$ $t>0$. The time argument $\mathcal{T}_{2\nu }(t),t>0$, is itself a random process whose distribution is related to the fractional diffusion equation. We also construct a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by the fractional Poisson process $\mathcal{N}_\nu.$ For this model we obtain the distributions of the random vector representing the position at time $t$, under the condition of a fixed number of events and in the unconditional case. For some specific values of $\nu\in(0,1]$ we show that the random position has a Brownian behavior (for $\nu =1/2$) or a cylindrical-wave structure (for $\nu =1$).

A fractional Poisson process in a model of dispersive charge transport in semiconductors

A Monte Carlo algorithm for the solution of an integro-differential equation with a frac- tional derivative is described, this method is based on appl ication of a model of a fractional Poisson process. The algorithm is used for the solution of the equati on of dispersive transport in disordered semiconductors. The transient photocurrent calculated by the Monte Carlo method is in good agree- ment with experimental data.

Fractional Poisson process (II)☆

Abstract In this paper, we propose a stochastic process W H ( t ) ( H ∈ ( 1 2 , 1 ) ) which we call fractional Poisson process. The process WH(t) is self-similar in wide sense, displays long range dependence, and has more fatter tail than Gaussian process. In addition, it converges to fractional Brownian motion in distribution.

Fractional Poisson process

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.