arXiv subjects

Barbara Martinucci

Publications and source records attributed to Barbara Martinucci.

At least 19 recordsLinked to original sources

arXiv2025

On a finite quasi birth-death process with catastrophes and its diffusion approximation

We study a multi-type Ehrenfest process modeled as a finite quasi-birth-death (QBD) process. We assume that the transitions are allowed only to the two adjacent levels of the same phase and are characterized by linear rates. The crucial element lies in the phase switching mechanism at the origin, which is governed by an irreducible stochastic matrix. The process evolution is interrupted by catastrophic events, whose occurrences are controlled by a Poisson process. Each catastrophe resets the system state to zero, initiating a new cycle of evolution until the next resetting event. We conduct a comprehensive analysis, addressing both the transient and long-term behavior of this process. Furthermore, we derive a diffusive approximation, by proving its convergence to a reflected Ornstein-Uhlenbeck jump diffusion process.

math.PR↗

arXiv2025

Extinction and Persistence in a Stochastic Mpox Model with Hawkes-type Self-Excitation

We develop a stochastic human-rodent compartment model for Mpox transmission that combines diffusion noise with Hawkes self-exciting jumps in the human infection dynamics. Including Hawkes processes allows, for instance, to model the short but significant spikes in transmission happening after crowded events. For the coupled human-rodent system, we prove global existence, uniqueness and positivity of solutions, derive a basic reproduction number R_0 that guarantees almost sure extinction when R_0 < 1, and obtain explicit persistence-in-the-mean conditions for both infected rodents and humans, which define persistence thresholds for the joint dynamics. Numerical experiments show how clustered human transmission events, environmental variability and control measures shift these thresholds and shape the frequency and size of Mpox outbreaks.

math.DS↗

arXiv2024

Applications of the Quantile-Based Probabilistic Mean Value Theorem to Distorted Distributions

Distorted distributions were introduced in the context of actuarial science for several variety of insurance problems. In this paper we consider the quantile-based probabilistic mean value theorem given in Di Crescenzo et al. [4] and provide some applications based on distorted random variables. Specifically, we consider the cases when the underlying random variables satisfy the proportional hazard rate model and the proportional reversed hazard rate model. A setting based on random variables having the 'new better than used' property is also analyzed.

math.PR↗

arXiv2022

Continuous-time multi-type Ehrenfest model and related Ornstein-Uhlenbeck diffusion on a star graph

We deal with a continuous-time Ehrenfest model defined over an extended star graph, defined as a lattice formed by the integers of $d$ semiaxis joined at the origin. The dynamics on each ray are regulated by linear transition rates, whereas the switching among rays at the origin occurs according to a general stochastic matrix. We perform a detailed investigation of the transient and asymptotic behavior of this process. We also obtain a diffusive approximation of the considered model, which leads to an Ornstein-Uhlenbeck diffusion process over a domain formed by semiaxis joined at the origin, named spider. We show that the approximating process possesses a truncated Gaussian stationary density. Finally, the goodness of the approximation is discussed through comparison of stationary distributions, means and variances.

math.PR↗

arXiv2021

Telegraph process with elastic boundary at the origin

We investigate the one-dimensional telegraph random process in the presence of an elastic boundary at the origin. This process describes a finite-velocity random motion that alternates between two possible directions of motion (positive or negative). When the particle hits the origin, it is either absorbed, with probability $α$, or reflected upwards, with probability $1-α$. In the case of exponentially distributed random times between consecutive changes of direction, we obtain the distribution of the renewal cycles and of the absorption time at the origin. This investigation is performed both in the case of motion starting from the origin and non-zero initial state. We also study the probability law of the process within a renewal cycle.

math.PR↗

arXiv2021

Piecewise linear processes with Poisson-modulated exponential switching times

We consider the jump telegraph process when switching intensities depend on external shocks also accompanying with jumps. The incomplete financial market model based on this process is studied. The Esscher transform, which changes only unobservable parameters, is considered in detail. The financial market model based on this transform can price switching risks as well as jump risks of the model.

math.PR↗

arXiv2020

Asymptotic results for the absorption time of telegraph processes with elastic boundary at the origin

We consider a telegraph process with elastic boundary at the origin studied recently in the literature. It is a particular random motion with finite velocity which starts at $x\geq 0$, and its dynamics is determined by upward and downward switching rates $λ$ and $μ$, with $λ>μ$, and an absorption probability (at the origin) $α\in(0,1]$. Our aim is to study the asymptotic behavior of the absorption time at the origin with respect to two different scalings: $x\to\infty$ in the first case; $μ\to\infty$, with $λ=βμ$ for some $β>1$ and $x>0$, in the second case. We prove several large and moderate deviation results. We also present numerical estimates of $β$ based on an asymptotic Normality result for the case of the second scaling.

math.PR↗

arXiv2020

Some results on the telegraph process confined by two non-standard boundaries

We analyze the one-dimensional telegraph random process confined by two boundaries, 0 and $H>0$. The process experiences hard reflection at the boundaries (with random switching to full absorption). Namely, when the process hits the origin (the threshold $H$) it is either absorbed, with probability $α$, or reflected upwards (downwards), with probability $1-α$, for $0<α<1$. We provide various results on the expected values of the renewal cycles and of the absorption time. The adopted approach is based on the analysis of the first-crossing times of a suitable compound Poisson process through linear boundaries. Our analysis includes also some comparisons between suitable stopping times of the considered telegraph process and of the corresponding diffusion process obtained under the classical Kac's scaling conditions.

math.PR↗

arXiv2019

Random time-changes and asymptotic results for a class of continuous-time Markov chains on integers with alternating rates

We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subodinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in Di Crescenzo A., Macci C., Martinucci B. (2014).

math.PR↗

arXiv2019

Analysis of random walks on a hexagonal lattice

We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a 2-dimensional Brownian motion is also discussed. Furthermore, we obtain some results on its asymptotic behavior making use of large deviation theory. Finally, we investigate the first-passage-time problem of the random walk through a vertical straight-line. Under suitable symmetry assumptions we are able to determine the first-passage-time probabilities in a closed form, which deserve interest in applied fields.

math.PR↗

arXiv2016

A multispecies birth-death-immigration process and its diffusion approximation

We consider an extended birth-death-immigration process defined on a lattice formed by the integers of $d$ semiaxes joined at the origin. When the process reaches the origin, then it may jumps toward any semiaxis with the same rate. The dynamics on each ray evolves according to a one-dimensional linear birth-death process with immigration. We investigate the transient and asymptotic behavior of the process via its probability generating function. The stationary distribution, when existing, is a zero-modified negative binomial distribution. We also study a diffusive approximation of the process, which involves a diffusion process with linear drift and infinitesimal variance on each ray. It possesses a gamma-type transient density admitting a stationary limit. As a byproduct of our study, we obtain a closed form of the number of permutations with a fixed number of components, and a new series form of the polylogarithm function expressed in terms of the Gauss hypergeometric function.

math.PR↗

arXiv2016

A fractional counting process and its connection with the Poisson process

We consider a fractional counting process with jumps of amplitude $1,2,\ldots,k$, with $k\in \mathbb{N}$, whose probabilities satisfy a suitable system of fractional difference-differential equations. We obtain the moment generating function and the probability law of the resulting process in terms of generalized Mittag-Leffler functions. We also discuss two equivalent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When $k=2$ we also express the distribution of the first passage time of the fractional counting process in an integral form. Finally, we show that the ratios given by the powers of the fractional Poisson process and of the counting process over their means tend to 1 in probability.

math.PR↗

arXiv2015

Compound Poisson process with a Poisson subordinator

A compound Poisson process whose randomized time is an independent Poisson process is called compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first-crossing-time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing-time density and survival functions.

math.PR↗

arXiv2015

A review on symmetry properties of birth-death processes

In this paper we review some results on time-homogeneous birth-death processes. Specifically, for truncated birth-death processes with two absorbing or two reflecting endpoints, we recall the necessary and sufficient conditions on the transition rates such that the transition probabilities satisfy a spatial symmetry relation. The latter leads to simple expressions for first-passage-time densities and avoiding transition probabilities. This approach is thus thoroughly extended to the case of bilateral birth-death processes, even in the presence of catastrophes, and to the case of a two-dimensional birth-death process with constant rates.

math.PR↗

arXiv2014

A quantile-based probabilistic mean value theorem

For nonnegative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows to construct new distributions with support (0,1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the `expected reversed proportional shortfall order', and a new characterization of random lifetimes involving the reversed hazard rate function.

math.PR↗

arXiv2013

A generalized telegraph process with velocity driven by random trials

We consider a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions. At each epoch of the underlying counting process the new velocity is determined by the outcome of a random trial. Two schemes are taken into account: Bernoulli trials and classical Pólya urn trials. We investigate the probability law of the process and the mean of the velocity of the moving particle. We finally discuss two cases of interest: (i) the case of Bernoulli trials and intertimes having exponential distributions with linear rates (in which, interestingly, the process exhibits a logistic stationary density with non-zero mean), and (ii) the case of Pólya trials and intertimes having first Gamma and then exponential distributions with constant rates.

math.PR↗

arXiv2013

On a bilateral birth-death process with alternating rates

We consider a bilateral birth-death process characterized by a constant transition rate $λ$ from even states and a possibly different transition rate $μ$ from odd states. We determine the probability generating functions of the even and odd states, the transition probabilities, mean and variance of the process for arbitrary initial state. Some features of the birth-death process confined to the non-negative integers by a reflecting boundary in the zero-state are also analyzed. In particular, making use of a Laplace transform approach we obtain a series form of the transition probability from state 1 to the zero-state.

math.PR↗

arXiv2008

Random motion with gamma-distributed alternating velocities in biological modeling

Motivated by applications in mathematical biology concerning randomly alternating motion of micro-organisms, we analyze a generalized integrated telegraph process. The random times between consecutive velocity reversals are gamma-distributed, and perform an alternating renewal process. We obtain the probability law and the mean of the process.

math.PR↗