scispace - formally typeset
Search or ask a question

Showing papers by "Barbara Martinucci published in 2012"


Journal ArticleDOI
TL;DR: In this article, the authors studied the distribution of the location at time t of a particle moving U time units upwards, V time units downwards, and W time units of no movement (idle).
Abstract: In this paper we study the distribution of the location, at time t , of a particle moving U time units upwards, V time units downwards, and W time units of no movement (idle). These are repeated cyclically, according to independent alternating renewals. The distributions of U , V , and W are absolutely continuous. The velocities are v = +1 upwards, v = -1 downwards, and v = 0 during idle periods. Let Y + ( t ), Y − ( t ), and Y 0 ( t ) denote the total time in (0, t ) of movements upwards, downwards, and no movements, respectively. The exact distribution of Y + ( t ) is derived. We also obtain the probability law of X ( t ) = Y + ( t ) - Y − ( t ), which describes the particle's location at time t . Explicit formulae are derived for the cases of exponential distributions with equal rates, with different rates, and with linear rates (leading to damped processes).

13 citations


Journal ArticleDOI
TL;DR: In this article, 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 was considered and the probability generating functions of the even and odd states, the transition probabilities, mean and variance of the process for arbitrary initial state were derived.
Abstract: 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.

12 citations


Journal ArticleDOI
TL;DR: In this paper, the authors considered a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions, and investigated the probability law of the process and the mean of the velocity of the moving particle.
Abstract: 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 P\'olya 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 P\'olya trials and intertimes having first Gamma and then exponential distributions with constant rates.

12 citations


Book ChapterDOI
01 Jan 2012
TL;DR: In this paper, the authors considered a related stochastic process, whose trajectories have two alternating slopes, for which the random times between consecutive slope changes have exponential distribution with linearly increasing parameters.
Abstract: The geometric telegrapher’s process has been proposed in 2002 as a model to describe the dynamics of the price of risky assets. In this contribution we consider a related stochastic process, whose trajectories have two alternating slopes, for which the random times between consecutive slope changes have exponential distribution with linearly increasing parameters. This leads to a process characterized by a damped behavior. We study the main features of the transient probability law of the process, and of its stationary limit.

8 citations