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The first rendezvous time of Brownian motion and compound Poisson-type processes

Part of: Markov processes Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

D. Perry ,
W. Stadje  and
S. Zacks
D. Perry*
Affiliation:
University of Haifa
W. Stadje*
Affiliation:
University of Osnabrück
S. Zacks*
Affiliation:
Binghamton University
*
∗ Postal address: Department of Statistics, University of Haifa, 31905 Haifa, Israel
∗∗ Postal address: Department of Mathematics and Computer Science, University of Osnabrück, 49069 Osnabrück, Germany. Email address: wolfgang@mathematik.uni-osnabrueck.de
∗∗∗ Postal address: Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA
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Abstract

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

Keywords

Brownian motioncompound Poisson processrendezvous timeoverflowemptinessLaplace transform

MSC classification

Primary: 60J65: Brownian motion 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1059 - 1070
Copyright
Copyright © Applied Probability Trust 2004 

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