Abstract
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 $\nu =1$).
Highlights
IntroductionAttempts to construct fractional versions of the Poisson process have been undertaken by Repin and Saichev (2000), Jumarie (2001) and Laskin (2003)
Attempts to construct fractional versions of the Poisson process have been undertaken by Repin and Saichev (2000), Jumarie (2001) and Laskin (2003).In analogy with the well-known fractional Brownian motion, the idea of passing from the usual Poisson process to a possible fractional version has inspired the papers by Wang et al (2006)(2007)
We construct the first type of fractional Poisson process (which we will denote by ν (t), t > 0) by replacing, in the differential equations governing the distribution of the classical Poisson process, the standard derivatives by the fractional derivatives, defined in (1.3), i.e. in the Dzerbayshan-Caputo sense
Summary
Attempts to construct fractional versions of the Poisson process have been undertaken by Repin and Saichev (2000), Jumarie (2001) and Laskin (2003). For ν = 1, (1.1) coincides with the equation governing the homogeneous Poisson process, so that our results generalize the well-known distributions holding in the standard case. Where B α, β denotes a Beta function of parameters α, β and B(t), t > 0, is a standard Brownian motion, independent of (X (t), Y (t)) This means that the planar motion with a Brownian time can be regarded as a planar Brownian motion, whose volatility is itself random and possesses a Beta distribution depending on the number of changes of directions. As we will see below, for ν = 1, formulae (1.16) and (1.17) reduce to the corresponding results holding for the standard Poisson case This version of fractional Poisson process does not possess the property of independence of increments.
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