Exit, passage, and crossing times and overshoots for a Poisson compound process with an exponential component
Author:
T. Kadankova
Translated by:
O. I. Klesov
Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom 75 (2006).
Journal:
Theor. Probability and Math. Statist. 75 (2007), 2339
MSC (2000):
Primary 60J05, 60J10; Secondary 60J45
Published electronically:
January 23, 2008
MathSciNet review:
2321178
Fulltext PDF Free Access
Abstract 
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Abstract: Integral transforms of the joint distribution of the first exit time from an interval and the overshoot over the boundary at the exit time are found for a Poisson process with an exponentially distributed negative component. We obtain the distributions of the following functionals of the process on an exponentially distributed time interval: the supremum, infimum, and the value of the process, numbers of upcrossings and downcrossings, the number of passages into an interval and overshoots over a boundary of an interval.
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 A. V. Skorokhod, Random Processes with Independent Increments, ``Nauka'', Moscow, 1964; English transl., Kluwer Academic Publishers, Dordrecht, 1991. MR 1155400 (93a:60114)
 2.
 I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes, vol. 2, ``Nauka'', Moscow, 1973; English transl., SpringerVerlag, New YorkHeidelberg, 1975.MR 0341540 (49:6288); MR 0375463 (51:11656)
 3.
 K. Itô and H. McKean, Diffusion Processes and their Sample Paths, SpringerVerlag, BerlinHeidelbergNew York, 1965. MR 0199891 (33:8031)
 4.
 L. Takács, Combinatorial Methods in the Theory of Stochastic Processes, John Wiley, New YorkLondonSydney, 1967. MR 0217858 (36:947)
 5.
 D. J. Emery, Exit problem for a spectrally positive process, Adv. Appl. Prob. 5 (1973), 498520. MR 0341623 (49:6370)
 6.
 E. A. Pecherskiĭ, Some identities related to the exit of a random walk from a segment and a semiinterval, Teor. Veroyatnost. i Primenen. 19 (1974), no. 1, 104119; English transl. in Theory Probab. Appl. 19 (1974), no. 1, 106121.MR 0341619 (49:6366)
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 V. N. Suprun and V. M. Shurenkov, On the resolvent of a process with independent increments that is terminated at the time of exit to the negative halfline, Studies in the Theory of Random Processes, Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1975, pp. 170174. (Russian) MR 0440712 (55:13583)
 8.
 V. N. Suprun, The ruin problem and resolvent of a killed process with independent increments, Ukrain. Mat. Zh. 28 (1976), no. 1, 5361, 142; English transl. in Ukrainian Math. J. 28 (1977), no. 1, 3945. MR 0428476 (55:1497)
 9.
 V. M. Shurenkov, Limiting distribution of the exit time out of an expanding interval and the position at this moment of a process with independent increments and onesigned jumps, Theory Probab. Appl. 23 (1978), no. 2, 419425. MR 0518318 (58:24572)
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 12.
 N. S. Bratiĭchuk and D. V. Gusak, Boundary Problems for Processes with Independent Increments, ``Naukova Dumka'', Kiev, 1990. (Russian) MR 1070711 (91m:60139)
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 V. F. Kadankov and T. V. Kadankova, On the distribution of the first exit time from an interval and the value of the overjump across a boundary for processes with independent increments and random walks, Ukrain. Mat. Zh. 57 (2005), no. 10, 13591384; English transl. in Ukrainian Math. J. 57 (2005), no. 10, 15901620. MR 2219768 (2007b:60119)
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 V. F. Kadankov and T. V. Kadankova, On the distribution of the moment of the first exit time from an interval and the value of overjump through borders interval for the processes with independent increments and random walks, Random Oper. Stochastic Equations 13 (2005), no. 3, 219244. MR 2165322 (2007b:60118)
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 E. A. Pecherskiĭ and B. A. Rogozin, The joint distributions of random variables associated to fluctuations of a process with independent increments, Teor. Veroyatnost. Primenen. 14 (1969), no. 3, 431444; English transl. in Theory Probab. Appl. 14 (1969), no. 3, 410423. MR 0260005 (41:4634)
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 18.
 T. V. Kadankova, On the distribution of the number of the intersections of a fixed interval by the semicontinuous process with independent increments, Theory Stoch. Process. (2003), no. 12, 7381. MR 2079924 (2005e:60102)
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 T. V. Kadankova, On the joint distribution of the supremum, infimum, and the value of a semicontinuous process with independent increments, Teor. Imovir. Mat. Stat. 70 (2004), 5665; English transl. in Theory Probab. Math. Statist. 70 (2005), 6170. MR 2109824 (2005h:60138)
 20.
 T. V. Kadankova, Twoboundary problems for a random walk with negative geometric jumps, Teor. Imovir. Mat. Stat. 68 (2003), 6071; English transl. in Theory Probab. Math. Statist. 68 (2004), 5566. MR 2000395 (2004f:60105)
 21.
 V. F. Kadankov and T. V. Kadankova, Intersections of an interval by a process with independent increments, Theory Stoch. Process. 11(27) (2005), no. 12, 5468. MR 2327447
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 T. O. Androshchuk, Distribution of the number of intersections of a segment by a random walk and the Brownian motion, Theory Stoch. Process. 7(23) (2001), no. 34, 37.
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Additional Information
T. Kadankova
Affiliation:
Center for Statistics, Hasselt University, Agoralaan, 3590 Diepenbeek, Belgium
Email:
tetyana.kadankova@uhasselt.be
DOI:
http://dx.doi.org/10.1090/S0094900008007114
PII:
S 00949000(08)007114
Keywords:
Poisson process with an exponentially distributed negative component,
oneboundary functionals of a process,
exit times from an interval,
overshoot over a boundary,
supremum and infimum of the process,
crossing times for an interval
Received by editor(s):
September 6, 2005
Published electronically:
January 23, 2008
Article copyright:
© Copyright 2008
American Mathematical Society
