Theory of Probability and Mathematical Statistics

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ISSN 1547-7363(e) ISSN 0094-9000(p)

Versions of a compound Poisson process

Author(s): D. V. Gusak
Translated by: V. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, vipusk 69 (2003).
Journal: Theor. Probability and Math. Statist. No. 69 (2004), 27-38.
MSC (2000): Primary 60G50, 60J70; Secondary 60K10, 60K15
Posted: February 7, 2005
MathSciNet review: 2110902
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Abstract: We consider two versions of an oscillating compound Poisson process with reflections from two boundaries. The versions are constructed from an upper continuous compound Poisson process $\xi(t)$ and two functionals of it, namely the exit time from an interval and first upcrossing or downcrossing times from the upper or lower boundaries, respectively. The basic characteristics of the processes considered in the paper are given in terms of the potential and resolvent of the process $\xi(t)$ introduced earlier by V. S. Korolyuk.

References:

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3.: -, Compound Poisson processes with two boundary reflection, Ukr. Matem. Zh. 11 (2002), 1616-1625; English transl. in Ukrain. Math. J. 11 (2002). MR 2016791 (2004i:60066)
4.: V. S. Korolyuk, Boundary Problems for Compound Poisson Processes, ``Naukova Dumka'', Kiev, 1975. (Russian) MR 0402939 (53:6753)
5.: V. S. Korolyuk, N. S. Brati ${\u{\i}}\kern.15em$ chuk, and B. Pirdzhanov, Boundary Problems for Random Walks, ``Ylym'', Ashgabad, 1987. (Russian)
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7.: E. A. Pecherski ${\u{\i}}\kern.15em$ , Some identities related to the exit of a random walk from a segment and a half-interval, Teor. Veroyatnost. Primenen. 19 (1974), no. 1, 104-109; English transl. in Theory Probab. Appl. 19 (1975), no. 1. MR 0341619 (49:6366)

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Additional Information:

D. V. Gusak
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivs'ka Street 3, Kyiv 4, Ukraine
Email: random@imath.kiev.ua

DOI: 10.1090/S0094-9000-05-00611-3
PII: S 0094-9000(05)00611-3
Keywords: Compound Poisson process, resolvent and potential functions, first exit time from an interval, moment generating function of the exit time, reflections of a Poisson process from two boundaries, versions of the risk process
Received by editor(s): 16/SEP/2002
Posted: February 7, 2005
Copyright of article: Copyright 2005, American Mathematical Society

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