Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-09T19:46:17.266Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic velocity motions and processes with random time

Part of: Functions of several variables Stochastic processes

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 1028 - 1056
Copyright
Copyright © Applied Probability Trust 2010 

References

Allouba, H. (2002). Brownian-time processes: the PDE connection. II. And the corresponding Feynman–Kac formula. Trans. Amer. Math. Soc. 354, 4627–4637.Google Scholar
Beghin, L. and Orsingher, E. (2009). Fractional Poisson processes and related planar random motions. Electron. J. Prob. 14, 1790–1827.Google Scholar
Burdzy, K. (1993). Some path properties of iterated Brownian motion. In Seminar on Stochastic Processes, eds Cinlar, E., Chung, K. L. and Sharpe, M. J., Birkhäuser, Boston, MA, pp. 67–87.Google Scholar
Burdzy, K. and Khoshnevisan, D. (1998). Brownian motion in a Brownian crack. Ann. Appl. Prob. 8, 708–748.Google Scholar
DeBlassie, R. D. (2004). Iterated Brownian motion in an open set. Ann. Appl. Prob. 14, 1529–1558.Google Scholar
De Gregorio, A. and Iacus, S. M. (2008). Parametric estimation for standard and geometric telegraph process observed at discrete times. Statist. Infer. Stoch. Process. 11, 249–263.CrossRefGoogle Scholar
De Gregorio, A. and Orsingher, E. (2006). Some results on random flights. Sci. Math. Jpn. 64, 351–356.Google Scholar
Di Crescenzo, A. (2001). On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Prob. 33, 690–701.CrossRefGoogle Scholar
Di Crescenzo, A. and Pellerey, F. (2002). On prices' evolutions based on geometric telegrapher's process. Appl. Stoch. Models Business Industry 18, 171–184.CrossRefGoogle Scholar
Foong, S. K. and Kanno, S. (1994). Properties of the telegrapher's random process with or without a trap. Stoch. Process. Appl. 53, 147–173.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (1980). Table of Integrals, Series and Products. Academic Press, New York.Google Scholar
Hillen, T. and Stevens, A. (2000). Hyperbolic models for chemotaxis in 1-D. Nonlinear Anal. Real World Appl. 1, 409–433.CrossRefGoogle Scholar
Holmes, E. E., Lewis, M. A., Banks, J. E. and Veit, R. R. (1994). Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 75, 17–29.Google Scholar
Iacus, S. M. and Yoshida, N. (2009). Estimation for discretely observed telegraph process. Theory Prob. Math. Statist. 78, 37–47.Google Scholar
Kolesnik, A. D. (2006). A four-dimensional random motion at finite speed. J. Appl. Prob. 43, 1107–1118.CrossRefGoogle Scholar
Kolesnik, A. D. and Orsingher, E. (2005). A planar random motion with an infinite number of directions controlled by the damped wave equation. J. Appl. Prob. 42, 1168–1182.CrossRefGoogle Scholar
Khoshnevisan, D. and Lewis, T. M. (1996). The uniform modulus of continuity of iterated Brownian motion. J. Theoret. Prob. 9, 317–333.Google Scholar
Mazza, C. and Rullière, D. (2004). A link between wave governed random motions and ruin processes. Insurance Math. Econom. 35, 205–222.CrossRefGoogle Scholar
Nane, E. (2006). Iterated Brownian motion in bounded domains in R n . Stoch. Process. Appl. 116, 905–916.Google Scholar
Orsingher, E. (1990). Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws. Stoch. Process. Appl. 34, 49–66.Google Scholar
Orsingher, E. and Beghin, L. (2009). Fractional diffusion equations and processes with randomly varying time. Ann. Prob. 37, 206–249.CrossRefGoogle Scholar
Orsingher, E. and De Gregorio, A. (2007). Random flights in higher spaces. J. Theoret. Prob. 20, 769–806.CrossRefGoogle Scholar
Ratanov, N. (2007a). A Jump telegraph model for option pricing. Quant. Finance 7, 575–583.CrossRefGoogle Scholar
Ratanov, N. (2007b). Jump telegraph processes and financial markets with memory. J. Appl. Math. Stoch. Anal. 2007, 72326, 19 pp.Google Scholar
Stadje, W. (1987). The exact probability distribution of a two-dimensional random walk. J. Statist. Phys. 46, 207–216.Google Scholar
Stadje, W. (1989). Exact probability distributions for noncorrelated random walk models. J. Statist. Phys. 56, 415–435.Google Scholar
Stadje, W. and Zacks, S. (2004). Telegraph processes with random velocities. J. Appl. Prob. 41, 665–678.CrossRefGoogle Scholar
Zacks, S. (2004). Generalized integrated telegraph processes and the distribution of related stopping times. J. Appl. Prob. 41, 497–507.Google Scholar
Weiss, G. H. (2002). Some applications of persistent random walks and the telegrapher's equation. Physica A 311, 381–410.CrossRefGoogle Scholar