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

Part of: Special processes Markov processes Stochastic 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.)
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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References

Abundo, M. (2001). Some results about boundary crossing for Brownian motion. Ric. Mat. 50, 283–301.Google Scholar
Kella, O., and Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 396–403.CrossRefGoogle Scholar
Kella, O., Perry, D., and Stadje, W. (2003). A stochastic clearing model with a Brownian and a compound Poisson component. Prob. Eng. Inf. Sci. 17, 1–22.CrossRefGoogle Scholar
Lehmann, A. (1998). Boundary crossing probabilities of Poisson counting processes with general boundaries. In Advances in Stochastic Models for Reliability, Quality and Safety, eds Kahle, W. et al., Birkhäuser, Boston, MA, pp. 153–156.CrossRefGoogle Scholar
Lehmann, A. (2002). Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes. Adv. Appl. Prob. 34, 869–887.CrossRefGoogle Scholar
Novikov, A., Frishling, V., and Kordzakhia, N. (1999). Approximations of boundary crossing probabilities for a Brownian motion. J. Appl. Prob. 36, 1019–1030.CrossRefGoogle Scholar
Perry, D., and Stadje, W. (2001). Function space integration for annuities. Insurance Math. Econom. 29, 73–82.CrossRefGoogle Scholar
Perry, D., Stadje, W., and Zacks, S. (2002). Boundary crossing for the difference of two ordinary or compound Poisson processes. Ann. Operat. Res. 113, 119–132.CrossRefGoogle Scholar
Perry, D., Stadje, W., and Zacks, S. (2004). A two-sided first-exit problem for a compound Poisson process with a random upper boundary. To appear in Methodology Comput. Appl. Prob.Google Scholar
Pötzelberger, K., and Wang, L. (2001). Boundary crossing probability for Brownian motion. J. Appl. Prob. 38, 152–164.CrossRefGoogle Scholar
Siegmund, D. (1985). Sequential Analysis. Springer, New York.CrossRefGoogle Scholar
Teunen, M., and Goovaerts, M. (1994). Double boundary crossing result for the Brownian motion. Scand. Actuarial J. 139–150.CrossRefGoogle Scholar
Thorisson, H. (2000). Coupling, Stationarity and Regeneration. Springer, New York.CrossRefGoogle Scholar
Wang, L. and Pötzelberger, K. (1997). Boundary crossing probability for Brownian motion and general boundaries. J. Appl. Prob. 34, 54–65.CrossRefGoogle Scholar
Wolff, R. (1989). Stochastic Modeling and The Theory of Queues. Prentice Hall, Englewood Cliffs, NJ.Google Scholar