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), 23-39

MSC (2000):
Primary 60J05, 60J10; Secondary 60J45

DOI:
https://doi.org/10.1090/S0094-9000-08-00711-4

Published electronically:
January 23, 2008

MathSciNet review:
2321178

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**A. V. Skorohod,*Random processes with independent increments*, Mathematics and its Applications (Soviet Series), vol. 47, Kluwer Academic Publishers Group, Dordrecht, 1991. Translated from the second Russian edition by P. V. Malyshev. MR**1155400****2.**I. I. Gikhman and A. V. Skorokhod,*\cyr Teoriya sluchaĭnykh protsessov. Tom II*, Izdat. “Nauka”, Moscow, 1973 (Russian). MR**0341540**

Ĭ. Ī. Gīhman and A. V. Skorohod,*The theory of stochastic processes. II*, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by Samuel Kotz; Die Grundlehren der Mathematischen Wissenschaften, Band 218. MR**0375463****3.**Kiyoshi Itô and Henry P. McKean Jr.,*Diffusion processes and their sample paths*, Die Grundlehren der Mathematischen Wissenschaften, Band 125, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-New York, 1965. MR**0199891****4.**Lajos Takács,*Combinatorial methods in the theory of stochastic processes*, John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR**0217858****5.**D. J. Emery,*Exit problem for a spectrally positive process*, Advances in Appl. Probability**5**(1973), 498–520. MR**0341623**, https://doi.org/10.2307/1425831**6.**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****7.**V. N. Suprun and V. M. Šurenkov,*On the resolvent of a process with independent increments that is terminated at the time of exit to the negative half-line*, Studies in the theory of random processes (Russian), Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1976, pp. 170–174 (Russian). MR**0440712****8.**V. N. Suprun,*The ruin problem and the resolvent of a killed independent increment process*, Ukrain. Mat. Ž.**28**(1976), no. 1, 53–61, 142 (Russian). MR**0428476****9.**V. M. Šurenkov,*Limiting distributions of the exit time out of an expanding interval and of the position at this time for a process with independent increments and jumps of the same sign*, Teor. Verojatnost. i Primenen.**23**(1978), no. 2, 419–425 (Russian, with English summary). MR**0518318****10.**E. B. Dynkin,*\cyr Markovskie protsessy*, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR**0193670****11.**V. S. Koroljuk,*\cyr Graniqnye zadaqi dlj slo+nyh puassonovskih processov.*, Izdat. “Naukova Dumka”, Kiev, 1975 (Russian). MR**0402939****12.**N. S. Bratiĭchuk and D. V. Gusak,*\cyr Granichnye zadachi dlya protsessov s nezavisimymi prirashcheniyami*, “Naukova Dumka”, Kiev, 1990 (Russian). With an English summary. MR**1070711****13.**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*, Ukraïn. Mat. Zh.**57**(2005), no. 10, 1359–1384 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J.**57**(2005), no. 10, 1590–1620. MR**2219768**, https://doi.org/10.1007/s11253-006-0016-6**14.**V. F. Kadankov and T. V. Kadankova,*On the distribution of the moment of the first exittime from an interval and value of overjump through borders interval for the processes with independent increments and random walk*, Random Oper. Stochastic Equations**13**(2005), no. 3, 219–244. MR**2165322**, https://doi.org/10.1163/156939705774286056**15.**E. A. Pečerskiĭ and B. A. Rogozin,*The combined distributions of the random variables connected with the fluctuations of a process with independent increments*, Teor. Verojatnost. i Primenen.**14**(1969), 431–444 (Russian, with English summary). MR**0260005****16.**A. A. Borovkov,*\cyr Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya.*, Izdat. “Nauka”, Moscow, 1972 (Russian). MR**0315800****17.**V. F. Kadankov and T. V. Kadankova,*On the distribution of duration of stay in an interval of the semi-continuous process with independent increments*, Random Oper. Stochastic Equations**12**(2004), no. 4, 361–384. MR**2108190**, https://doi.org/10.1163/1569397042722355**18.**Tatiana V. Kadankova,*On the distribution of the number of the intersections of a fixed interval by the semi-continuous process with independent increments*, Theory Stoch. Process.**9**(2003), no. 1-2, 73–81. MR**2079924****19.**T. V. Kadankova,*On the joint distribution of the supremum, infimum, and the value of a semicontinuous process with independent increments*, Teor. Ĭmovīr. Mat. Stat.**70**(2004), 54–62 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**70**(2005), 61–70. MR**2109824**, https://doi.org/10.1090/S0094-9000-05-00631-9**20.**T. V. Kadankova,*Two-boundary problems for a random walk with geometrically distributed negative jumps*, Teor. Ĭmovīr. Mat. Stat.**68**(2003), 49–60 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**68**(2004), 55–66. MR**2000395**, https://doi.org/10.1090/S0094-9000-04-00604-0**21.**V. F. Kadankov and T. V. Kadankova,*Intersections of an interval by a process with independent increments*, Theory Stoch. Process.**11**(2005), no. 1-2, 54–68. MR**2327447****22.**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. 3-4, 3-7.**23.**V. A. Ditkin and A. P. Prudnikov,*Integral transforms and operational calculus*, Translated by D. E. Brown. English translation edited by Ian N. Sneddon, Pergamon Press, Oxford-Edinburgh-New York, 1965. MR**0196422**

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Additional Information

**T. Kadankova**

Affiliation:
Center for Statistics, Hasselt University, Agoralaan, 3590 Diepenbeek, Belgium

Email:
tetyana.kadankova@uhasselt.be

DOI:
https://doi.org/10.1090/S0094-9000-08-00711-4

Keywords:
Poisson process with an exponentially distributed negative component,
one-boundary 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