Geometric Stable processes and related fractional differential equations

We are interested in the differential equations satisfied by the density of the Geometric Stable processes $\mathcal{G}_{\alpha}^{\beta}=\left\{\mathcal{G}_{\alpha}^{\beta}(t);t\geq 0\right\}$, with stability \ index $$ and asymmetry parameter $\beta \in \lbrack -1,1]$, both in the univariate and in the multivariate cases. We resort to their representation as compositions of stable processes with an independent Gamma subordinator. As a preliminary result, we prove that the latter is governed by a differential equation expressed by means of the shift operator. As a consequence, we obtain the space-fractional equation satisfied by the density of $\mathcal{G}_{\alpha}^{\beta}.$ For some particular values of $$ and $\beta ,$ we get some interesting results linked to well-known processes, such as the Variance Gamma process and the first passage time of the Brownian motion.Comment: 12 page

Long-memory Gaussian processes governed by generalized Fokker-Planck equations

It is well-known that the transition function of the Ornstein-Uhlenbeck process solves the Fokker-Planck equation. This standard setting has been recently generalized in different directions, for example, by considering the so-called $\alpha$-stable driven Ornstein-Uhlenbeck, or by time-changing the original process with an inverse stable subordinator. In both cases, the corresponding partial differential equations involve fractional derivatives (of Riesz and Riemann-Liouville types, respectively) and the solution is not Gaussian. We consider here a new model, which cannot be expressed by a random time-change of the original process: we start by a Fokker-Planck equation (in Fourier space) with the time-derivative replaced by a new fractional differential operator. The resulting process is Gaussian and, in the stationary case, exhibits a long-range dependence. Moreover, we consider further extensions, by means of the so-called convolution-type derivative.Comment: 24, accepted for publicatio

Multivariate fractional Poisson processes and compound sums

In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (nonfractional) Poisson processes. In some cases we also consider compound processes. We obtain some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes

Large deviations for fractional Poisson processes

We prove large deviation principles for two versions of fractional Poisson processes. Firstly we consider the main version which is a renewal process; we also present large deviation estimates for the ruin probabilities of an insurance model with constant premium rate, i.i.d. light tail claim sizes, and a fractional Poisson claim number process. We conclude with the alternative version where all the random variables are weighted Poisson distributed. Keywords: Mittag Leffler function; renewal process; random time ch

Fractional diffusion equations and processes with randomly varying time

In this paper the solutions $u_{\nu}=u_{\nu}(x,t)$ to fractional diffusion equations of order $0<\nu \leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $\nu =\frac{1}{2^n}$, $n\geq 1,$ we show that the solutions $u_{{1/2^n}}$ correspond to the distribution of the $n$-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order $\nu =\frac{2}{3^n}$, $n\geq 1,$ is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that $u_{\nu}$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $u_{\nu}$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.Comment: Published in at http://dx.doi.org/10.1214/08-AOP401 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The distribution of the local time for "pseudo-processes" and its connections with fractional diffusion equations

We prove that the pseudoprocesses governed by heat-type equations of order $n\geq2$ have a local time in zero (denoted by $L_{0}^{n}(t)$) whose distribution coincides with the folded fundamental solution of a fractional diffusion equation of order $2(n-1)/n,n\geq2$: The distribution of $L_{0}^{n}(t)$ is also expressed in terms of stable laws of order $n/(n-1)$ and their form is analyzed. Furthermore, it is proved that the distribution of $L_{0}^{n}(t)$ is connected with a wave equation as $n\rightarrow\infty$. The distribution of the local time in zero for the pseudoprocess related to the Myiamoto’s equation is also derived and examined together with the corresponding telegraph-type fractional equation

Poisson-type processes governed by fractional and higher-order recursive differential equations

We consider some fractional extensions of the recursive differential equation governing the Poisson process, by introducing combinations of different fractional time-derivatives. We show that the so-called "Generalized Mittag-Leffler functions" (introduced by Prabhakar [20]) arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the intearrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for t tending to infinite. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter fractional parameter varying in the interval (0,1). For integer values of the parameter, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.Comment: 37 pages, 1 figur

Multivariate fractional Poisson processes and compound sums

In this paper we present multivariate space-time fractional Poisson processes by considering common random time-changes of a (finite-dimensional) vector of independent classical (non-fractional) Poisson processes. In some cases we also consider compound processes. We obtain some equations in terms of some suitable fractional derivatives and fractional difference operators, which provides the extension of known equations for the univariate processes.Comment: 19 pages Keywords: conditional independence, Fox-Wright function, fractional differential equations, random time-chang