Versions of a compound Poisson process
Author:
D. V. Gusak
Translated by:
V. Semenov
Journal:
Theor. Probability and Math. Statist. 69 (2004), 27-38
MSC (2000):
Primary 60G50, 60J70; Secondary 60K10, 60K15
DOI:
https://doi.org/10.1090/S0094-9000-05-00611-3
Published electronically:
February 7, 2005
MathSciNet review:
2110902
Full-text PDF Free Access
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Additional Information
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
- D. V. Gusak, On the modified risk process with reflection, Theory Stoch. Process. 3 (19) (1997), no. 1–2, 197–207.
- D. V. Gusak, On modifications of risk processes, Teor. Ĭmovīr. Mat. Stat. 56 (1997), 87–95 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 56 (1998), 87–95. MR 1791857
- D. V. Gusak, Compound Poisson processes with two-sided reflection, Ukraïn. Mat. Zh. 54 (2002), no. 12, 1616–1625 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 54 (2002), no. 12, 1958–1970. MR 2016791, DOI https://doi.org/10.1023/A%3A1024069130470
- V. S. Koroljuk, Graniqnye zadaqi dlj slo+nyh puassonovskih processov, Izdat. “Naukova Dumka”, Kiev, 1975 (Russian). MR 0402939
- V. S. Korolyuk, N. S. Bratiĭchuk, and B. Pirdzhanov, Boundary Problems for Random Walks, “Ylym”, Ashgabad, 1987. (Russian)
- N. S. Bratiĭchuk and D. V. Gusak, Granichnye zadachi dlya protsessov s nezavisimymi prirashcheniyami, “Naukova Dumka”, Kiev, 1990 (Russian). With an English summary. MR 1070711
- E. A. Pečerskiĭ, Certain identities that are connected with the exit of a random walk from a segment and from a half-interval, Teor. Verojatnost. i Primenen. 19 (1974), 104–119 (Russian, with English summary). MR 0341619
References
- D. V. Gusak, On the modified risk process with reflection, Theory Stoch. Process. 3 (19) (1997), no. 1–2, 197–207.
- ---, On modifications of risk processes, Teor. Imovirnost. Matem. Statist. 56 (1998), 87–95; English transl. in Theory Probab. Math. Statist. 56 (1999), 87–95. MR 1791857 (2002m:60080)
- ---, 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)
- V. S. Korolyuk, Boundary Problems for Compound Poisson Processes, “Naukova Dumka”, Kiev, 1975. (Russian) MR 0402939 (53:6753)
- V. S. Korolyuk, N. S. Bratiĭchuk, and B. Pirdzhanov, Boundary Problems for Random Walks, “Ylym”, Ashgabad, 1987. (Russian)
- N. S. Bratiĭchuk and D. V. Gusak, Limit Problems for Processes with Independent Increments, “Naukova Dumka”, Kiev, 1990. (Russian) MR 1070711 (91m:60139)
- E. A. Pecherskiĭ, 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
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):
September 16, 2002
Published electronically:
February 7, 2005
Article copyright:
© Copyright 2005
American Mathematical Society
