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Stochastic velocity motions and processes with random time

Part of: Stochastic processes Functions of several variables

Published online by Cambridge University Press:  01 July 2016

Alessandro De Gregorio
Alessandro De Gregorio*
Affiliation:
Sapienza University of Rome
*
∗ Postal address: Department of Statistical Science, Sapienza University of Rome, Piazzale Aldo Moro, 5 - 00185, Rome, Italy. Email address: alessandro.degregorio@uniroma1.it
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Abstract

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The aim of this paper is to analyze a class of random processes which models the motion of a particle on the real line with random velocity and subject to the action of friction. The speed randomly changes when a Poissonian event occurs. We study the characteristic and moment generating functions of the position reached by the particle at time t > 0. We are able to derive the explicit probability distributions in a few cases. The moments are also widely analyzed. For the random motions having an explicit density law, further interesting probabilistic interpretations emerge if we consider randomly varying time. Essentially, we consider two different types of random time, namely Bessel and gamma times, which contain, as particular cases, some important probability distributions (e.g. Gaussian, exponential). For the random processes built by means of these compositions, we derive the probability distributions for a fixed number of Poisson events. Some remarks on possible extensions to random motions in higher spaces are proposed. We focus our attention on the persistent planar random motion.

Keywords

Bessel processgamma processiterated Brownian motionStruve functionLaplace distributionrandom flightrandom timetelegraph process

MSC classification

Primary: 60G07: General theory of processes 26B15: Integration
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 1028 - 1056
Copyright
Copyright © Applied Probability Trust 2010 

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