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A Simple Model for Random Oscillations

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

F. Papangelou
F. Papangelou*
Affiliation:
University of Manchester
*
∗Postal address: School of Mathematics, University of Manchester, Manchester, M13 9PL, UK. Email address: fredos.papangelou@manchester.ac.uk
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Abstract

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A simple model for a randomly oscillating variable is suggested, which is a variant of the two-state random velocity model. As in the latter model, the variable keeps rising or falling with constant velocity for some time before randomly reversing its direction. In contrast however, its propensity to reverse depends on its current value and it is for this desirable feature that the model is proposed here. This feature has two implications: (a) neither the changing variable nor its velocity is Markovian, although the joint process is, and (b) the linear differential equations arising in the case of our model do not have constant coefficients. The results given in this paper are meant to illustrate the straightforward nature of some of the calculations involved and to highlight the relationship with one-dimensional diffusions.

Keywords

Random oscillationrandom evolutioninertiahazard fieldMarkov processgeneratorstationary distributionrenewallevel crossingrecurrence timeexit distributionexit timediffusionscale and speed measure

MSC classification

Secondary: 60K40: Other physical applications of random processes 60J25: Continuous-time Markov processes on general state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1164 - 1173
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Dynkin, E. B. (1965). Markov Processes, Vol. I. Springer, Berlin.Google Scholar
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[3] Klebaner, F. C. (1998). Introduction to Stochastic Calculus with Applications. Imperial College Press, London.Google Scholar
[4] Pinsky, M. A. (1991). Lectures on Random Evolution. World Scientific, River Edge, NJ.Google Scholar