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UPPER FIRST-EXIT TIMES OF COMPOUND POISSON PROCESSES REVISITED

Published online by Cambridge University Press:  24 September 2003

W. Stadje  and
S. Zacks
W. Stadje
Affiliation:
Fachbereich Mathematik/Informatik, University of Osnabrück, 49069 Osnabrück, Germany, E-mail: wolfgang@mathematik.uni-osnabrueck.de
S. Zacks
Affiliation:
Department of Mathematical Sciences, Binghamton University, Binghamton, New York 13902-6000, E-mail: shelly@math.binghamton.edu
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Abstract

For a compound Poisson process (CPP) with only positive jumps, an elegant formula connects the density of the hitting time for a lower straight line with that of the process itself at time t, h(x; t), considered as a function of time and position jointly. We prove an analogous (albeit more complicated) result for the first time the CPP crosses an upper straight line. We also consider the conditional density of the CPP at time t, given that the upper line has not been reached before t. Finally, it is shown how to compute certain moment integrals of h.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 17 , Issue 4 , October 2003 , pp. 459 - 465
Copyright
© 2003 Cambridge University Press

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