Probability Law and Flow Function of Brownian Motion Driven by a Generalized Telegraph Process

Antonio Di Crescenzo () and Shelemyahu Zacks ()
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Antonio Di Crescenzo: Università di Salerno
Shelemyahu Zacks: Binghamton University

Methodology and Computing in Applied Probability, 2015, vol. 17, issue 3, 761-780

Abstract: Abstract We consider a standard Brownian motion whose drift alternates randomly between a positive and a negative value, according to a generalized telegraph process. We first investigate the distribution of the occupation time, i.e. the fraction of time when the motion moves with positive drift. This allows to obtain explicitly the probability law and the flow function of the random motion. We discuss three special cases when the times separating consecutive drift changes have (i) exponential distribution with constant rates, (ii) Erlang distribution, and (iii) exponential distribution with linear rates. In conclusion, in view of an application in environmental sciences we evaluate the density of a Wiener process with infinitesimal moments alternating at inverse Gaussian distributed random times.

Keywords: Standard Brownian motion; Alternating drift; Alternating counting process; Exponential random times; Erlang random times; Modified Bessel function; Two-index pseudo-Bessel function; 60J65; 60K15 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (7)

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DOI: 10.1007/s11009-013-9392-1

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