Exit time functionals for integer-valued Poisson processes
Author:
D. V. Gusak
Translated by:
V. Semenov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 68 (2003).
Journal:
Theor. Probability and Math. Statist. 68 (2004), 27-39
MSC (2000):
Primary 60G50, 60J70
DOI:
https://doi.org/10.1090/S0094-9000-04-00603-9
Published electronically:
June 10, 2004
MathSciNet review:
2000392
Full-text PDF Free Access
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Abstract: The joint distribution of all exit time functionals is studied in this paper for a fixed level $x$ and integer-valued compound Poisson processes. An exact formula for the distributions of these functionals is obtained in the case of semicontinuous processes. Limit relations are obtained for the distributions of the exit time functionals for $x=0$ or as $x\to \infty$.
- D. V. Gusak, The joint distribution of the time and magnitude of the first overshoot for homogeneous processes with independent increments, Teor. Verojatnost. i Primenen. 14 (1969), 15–23 (Russian, with English summary). MR 0245083
- D. V. Gusak, Distribution of overshoot functionals of a semicontinuous homogeneous process with independent increments, Ukraïn. Mat. Zh. 54 (2002), no. 3, 303–322 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 54 (2002), no. 3, 371–397. MR 1952790, DOI https://doi.org/10.1023/A%3A1020557215381
- 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
- D. V. Gusak and A. I. Tureniyazova, Distribution of some functionals for lattice Poisson processes on a Markov chain, Asymptotic methods in the investigation of stochastic models (Russian), Acad. Sci. Ukrain. SSR, Inst. Math., Kiev, 1987, pp. 21–27, 143 (Russian). MR 943353
- D. V. Gusak and A. I. Tureniyazova, On lattice semicontinuous Poisson processes on a Markov chain, Ukrain. Mat. Zh. 39 (1987), no. 6, 707–711, 813 (Russian). MR 925795
6 D. V. Gusak, On a generalized semicontinuous integer-valued Poisson process with reflection, Teoriya Imovir. ta Matem. Statist. 59 (1998), 41–46; English transl., Theor. Probability and Math. Statist. 59 (1999), 41–46.
7 ---, The factorization method in boundary problems for homogeneous processes with independent increments, Distribution of Some Functionals for Processes with Independent Increments, Preprint 85-43, Institute of Mathematics, Academy of Sciences of Ukrainian SSR, Kiev, 1985, pp. 3–42. (Russian)
1 D. V. Gusak, On the joint distribution of the first exit time and exit value for homogeneous processes with independent increments, Teor. Veroyatnost. i Primenen. 14 (1969), no. 1, 15–23; English transl. in Theor. Probab. Appl. 14 (1970), no. 1.
2 ---, Distribution of overjump functionals of a homogeneous process with independent increments, Ukrain. Mat. Zh. 54 (2002), no. 3, 303–322; English transl., Ukrain. Math. J. 54 (2003), no. 3, 371–397.
3 N. S. Bratiĭchuk and D. V. Gusak, Boundary Problems for Processes with Independent Increments, “Naukova Dumka”, Kiev, 1990. (Russian)
4 D. V. Gusak and A. I. Tureniyazova, The distribution of some limit functionals for lattice Poisson processes defined on a Markov chain, Asymptotic Methods in Studies of Stochastic Models, Institute of Mathematics, Academy of Sciences of Ukrainian SSR, Kiev, 1987, pp. 21–27. (Russian)
5 ---, On lattice semicontinuous processes defined on a Markov chain, Ukrain. Mat. Zh. 39 (1987), no. 6, 707–711; English transl. in Ukrain. Math. J. 39 (1988), no. 6.
6 D. V. Gusak, On a generalized semicontinuous integer-valued Poisson process with reflection, Teoriya Imovir. ta Matem. Statist. 59 (1998), 41–46; English transl., Theor. Probability and Math. Statist. 59 (1999), 41–46.
7 ---, The factorization method in boundary problems for homogeneous processes with independent increments, Distribution of Some Functionals for Processes with Independent Increments, Preprint 85-43, Institute of Mathematics, Academy of Sciences of Ukrainian SSR, Kiev, 1985, pp. 3–42. (Russian)
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Additional Information
D. V. Gusak
Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivs’ka Street 3, Kyiv 01601, Ukraine
Email:
random@imath.kiev.ua
Received by editor(s):
February 18, 2002
Published electronically:
June 10, 2004
Article copyright:
© Copyright 2004
American Mathematical Society
