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Exit time functionals for integer-valued Poisson processes

Author(s): D. V. Gusak
Translated by: V. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, vipusk 68 (2003).
Journal: Theor. Probability and Math. Statist. No. 68 (2004), 27-39.
MSC (2000): Primary 60G50, 60J70
Posted: June 10, 2004
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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 $x\to\infty$.

References:

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. MR 39:6395

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-, 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. MR 2003j:60067

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N. S. Brati{\u{\i}}\kern.15emchuk and D. V. Gusak, Boundary Problems for Processes with Independent Increments, ``Naukova Dumka'', Kiev, 1990. (Russian) MR 91m:60139

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) MR 90f:60132

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-, 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. MR 89a:60172

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

DOI: 10.1090/S0094-9000-04-00603-9
PII: S 0094-9000(04)00603-9
Received by editor(s): 18/FEB/2002
Posted: June 10, 2004
Copyright of article: Copyright 2004, American Mathematical Society

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