arXiv subjects

Luisa Beghin

Publications and source records attributed to Luisa Beghin.

36 records · Page 2Linked to original sources

arXiv2019

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 $α$-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.

math.PR↗

arXiv2018

Pseudo-differential operators and related additive geometric stable processes

Additive processes are obtained from Lévy ones by relaxing the condition of stationary increments, hence they are spatially (but not temporally) homogeneous. By analogy with the case of time-homogeneous Markov processes, one can define an infinitesimal generator, which is, of course, a time-dependent operator. Additive versions of stable and Gamma processes have been considered in the literature. We introduce here time-inhomogeneous generalizations of the well-known geometric stable process, defined by means of time-dependent versions of fractional pseudo-differential operators of logarithmic type. The local Lévy measures are expressed in terms of Mittag-Leffler functions or $H$-functions with time-dependent parameters.

math.PR↗

arXiv2016

Asymptotic results for a multivariate version of the alternative fractional Poisson process

A multivariate fractional Poisson process was recently defined in Beghin and Macci (2016) by considering a common independent random time change for a finite dimensional vector of independent (non-fractional) Poisson processes; moreover it was proved that, for each fixed $t\geq 0$, it has a suitable multinomial conditional distribution of the components given their sum. In this paper we consider another multivariate process $\{\underline{M}^ν(t)=(M_1^ν(t),\ldots,M_m^ν(t)):t\geq 0\}$ with the same conditional distributions of the components given their sums, and different marginal distributions of the sums; more precisely we assume that the one-dimensional marginal distributions of the process $\left\{\sum_{i=1}^mM_i^ν(t):t\geq0\right\}$ coincide with the ones of the alternative fractional (univariate) Poisson process in Beghin and Macci (2013). We present large deviation results for $\{\underline{M}^ν(t)=(M_1^ν(t),\ldots,M_m^ν(t)):t\geq 0\}$, and this generalizes the result in Beghin and Macci (2013) concerning the univariate case. We also study moderate deviations and we present some statistical applications concerning the estimation of the fractional parameter $ν$.

math.PR↗

arXiv2016

Time-inhomogeneous fractional Poisson processes defined by the multistable subordinator

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.

math.PR↗

arXiv2016

Fractional diffusion-type equations with exponential and logarithmic differential operators

We deal with some extensions of the space-fractional diffusion equation, which is satisfied by the density of a stable process (see Mainardi, Luchko, Pagnini (2001)): the first equation considered here is obtained by adding an exponential differential operator expressed in terms of the Riesz-Feller derivative. We prove that this produces a random additional term in the time-argument of the corresponding stable process, which is represented by the so-called Poisson process with drift. Analogously, if we add, to the space-fractional diffusion equation, a logarithmic differential operator involving the Riesz-derivative, we obtain, as a solution, the transition semigroup of a stable process subordinated by an independent gamma subordinator with drift. Finally, we show that a non-linear extension of the space-fractional diffusion equation is satisfied by the transition density of the process obtained by time-changing the stable process with an independent linear birth process with drift.

math.PR↗

arXiv2015

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.

math.PR↗

arXiv2014

Correlated fractional counting processes on a finite time interval

We present some correlated fractional counting processes on a finite time interval. This will be done by considering a slight generalization of the processes in Borges et al. (2012). The main case concerns a class of space-time fractional Poisson processes and, when the correlation parameter is equal to zero, the univariate distributions coincide with the ones of the space-time fractional Poisson process in Orsingher and Polito (2012). On the other hand, when we consider the time fractional Poisson process, the multivariate finite dimensional distributions are different from the ones presented for the renewal process in Politi et al. (2011). Another case concerns a class of fractional negative binomial processes.

math.PR↗

arXiv2014

Fractional Poisson process with random drift

We study the connection between PDEs and Lévy 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 Frobenious-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.

math.PR↗

arXiv2013

Fractional Gamma process and fractional Gamma-subordinated processes

We define and study fractional versions of the well-known Gamma subordinator $Γ:=\{Γ(t),$ $t\geq 0\},$ which are obtained by time-changing $% Γ$ by means of an independent stable subordinator or its inverse. Their densities are proved to satisfy differential equations expressed in terms of fractional versions of the shift operator (with fractional parameter greater or less than one, in the two cases). As a consequence, the fractional generalization of some Gamma subordinated processes (i.e. the Variance Gamma, the Geometric Stable and the Negative Binomial) are introduced and the corresponding fractional differential equations are obtained.

math.PR↗

arXiv2013

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}_α^β=\left\{\mathcal{G}_α^β(t);t\geq 0\right\} $, with stability \ index $% α\in (0,2]$ and asymmetry parameter $β\in \lbrack -1,1]$, 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 $\mathcal{G}_α^β.$ For some particular values of $% α$ and $β,$ 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.

math.PR↗

arXiv2013

Fractional discrete processes: compound and mixed Poisson representations

We consider two fractional versions of a family of nonnegative integer valued processes. We prove that their probability mass functions solve fractional Kolmogorov forward equations, and we show the overdispersion of these processes. As particular examples in this family, we can define fractional versions of some processes in the literature as the Polya-Aeppli, the Poisson Inverse Gaussian and the Negative Binomial. We also define and study some more general fractional versions with two fractional parameters.

math.PR↗

arXiv2012

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 cha

math.PR↗

arXiv2011

Fractional relaxation equations and Brownian crossing probabilities of a random boundary

We analyze here different forms of fractional relaxation equations of order ν\in(0,1) and we derive their solutions both in analytical and in probabilistic forms. In particular we show that these solutions can be expressed as crossing probabilities of random boundaries by various types of stochastic processes, which are all related to the Brownian motion B. In the special case ν=1/2, the fractional relaxation is proved to coincide with Pr{sup_{0\leqs\leqt} B(s)<U}, for an exponential boundary U. When we generalize the distributions of the random boundary, passing from the exponential to the Gamma density, we obtain more and more complicated fractional equations.

math.PR↗

arXiv2011

Random-time processes governed by differential equations of fractional distributed order

We analyze here different types of fractional differential equations, under the assumption that their fractional order $ν\in (0,1] $ is random\ with probability density $n(ν).$ We start by considering the fractional extension of the recursive equation governing the homogeneous Poisson process $N(t),t>0.$\ We prove that, for a particular (discrete) choice of $n(ν)$, it leads to a process with random time, defined as $N(% \widetilde{\mathcal{T}}_{ν_{1,}ν_{2}}(t)),t>0.$ The distribution of the random time argument $\widetilde{\mathcal{T}}_{ν_{1,}ν_{2}}(t)$ can be expressed, for any fixed $t$, in terms of convolutions of stable-laws. The new process $N(\widetilde{\mathcal{T}}_{ν_{1,}ν_{2}})$ is itself a renewal and can be shown to be a Cox process. Moreover we prove that the survival probability of $N(\widetilde{\mathcal{T}}_{ν_{1,}ν_{2}})$, as well as its probability generating function, are solution to the so-called fractional relaxation equation of distributed order (see \cite{Vib}%). In view of the previous results it is natural to consider diffusion-type fractional equations of distributed order. We present here an approach to their solutions in terms of composition of the Brownian motion $B(t),t>0$ with the random time $\widetilde{\mathcal{T}}_{ν_{1,}ν_{2}}$. We thus provide an alternative to the constructions presented in Mainardi and Pagnini \cite{mapagn} and in Chechkin et al. \cite{che1}, at least in the double-order case.

math.PR↗

arXiv2011

Fractional diffusion equations and processes with randomly varying time

In this paper the solutions $u_ν=u_ν(x,t)$ to fractional diffusion equations of order $0<ν\leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $ν=\frac{1}{2^n}$, $n\geq 1,$ 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 $ν=\frac{2}{3^n}$, $n\geq 1,$ 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 $u_ν$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $u_ν$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

math.PR↗

arXiv2010

Equations of Mathematical Physics and Compositions of Brownian and Cauchy processes

We consider different types of processes obtained by composing Brownian motion $B(t)$, fractional Brownian motion $B_{H}(t)$ and Cauchy processes $% C(t)$ in different manners. We study also multidimensional iterated processes in $\mathbb{R}^{d},$ like, for example, $\left( B_{1}(|C(t)|),...,B_{d}(|C(t)|)\right) $ and $\left( C_{1}(|C(t)|),...,C_{d}(|C(t)|)\right) ,$ deriving the corresponding partial differential equations satisfied by their joint distribution. We show that many important partial differential equations, like wave equation, equation of vibration of rods, higher-order heat equation, are satisfied by the laws of the iterated processes considered in the work. Similarly we prove that some processes like $% C(|B_{1}(|B_{2}(...|B_{n+1}(t)|...)|)|)$ are governed by fractional diffusion equations.

math.PR↗

arXiv2009

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.

math.PR↗

arXiv2007

Pseudoprocesses governed by higher-order fractional differential equations

We study here a heat-type differential equation of order n greater than two, in the case where the time-derivative is supposed to be fractional. The corresponding solution can be described as the transition function of a pseudoprocess (coinciding with the one governed by the standard, non-fractional, equation) with a time argument T which is itself random. The distribution of T is presented together with some features of the solution (such as analytic expressions for its moments).

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