Geodesic
On the distribution of the first exit time and overshoot in a two-sided boundary crossing problem
Matematičeskie trudy, Tome 15 (2012) no. 1, pp. 109-119 
V. I. Lotov; A. S. Tarasenko. On the distribution of the first exit time and overshoot in a two-sided boundary crossing problem. Matematičeskie trudy, Tome 15 (2012) no. 1, pp. 109-119. http://geodesic.mathdoc.fr/item/MT_2012_15_1_a5/
@article{MT_2012_15_1_a5,
     author = {V. I. Lotov and A. S. Tarasenko},
     title = {On the distribution of the first exit time and overshoot in a~two-sided boundary crossing problem},
     journal = {Matemati\v{c}eskie trudy},
     pages = {109--119},
     year = {2012},
     volume = {15},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2012_15_1_a5/}
}
TY  - JOUR
AU  - V. I. Lotov
AU  - A. S. Tarasenko
TI  - On the distribution of the first exit time and overshoot in a two-sided boundary crossing problem
JO  - Matematičeskie trudy
PY  - 2012
SP  - 109
EP  - 119
VL  - 15
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MT_2012_15_1_a5/
LA  - ru
ID  - MT_2012_15_1_a5
ER  - 
%0 Journal Article
%A V. I. Lotov
%A A. S. Tarasenko
%T On the distribution of the first exit time and overshoot in a two-sided boundary crossing problem
%J Matematičeskie trudy
%D 2012
%P 109-119
%V 15
%N 1
%U http://geodesic.mathdoc.fr/item/MT_2012_15_1_a5/
%G ru
%F MT_2012_15_1_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a random walk generated by a sequence of independent identically distributed random variables. We assume that the distribution function of a jump of the random walk equals an exponential polynomial on the negative half-axis. For double transforms of the joint distribution of the first exit time from an interval and overshoot, we obtain explicit expressions depending on finitely many parameters that, in turn, we can derive from the system of linear equations. The principal difference of the present article from similar results in this direction is the rejection of using factorization components and projection operators connected with them.

[1] Borovkov A. A., “O faktorizatsionnykh tozhdestvakh i svoistvakh raspredeleniya supremuma posledovatelnykh summ”, TVP, 15:3 (1970), 377–418 | MR | Zbl

[2] Lotov V. I., “Asimptoticheskii analiz raspredelenii v dvugranichnykh zadachakh. I”, TVP, 24:3 (1979), 475–485 | MR | Zbl

[3] Lotov V. I., “Ob asimptotike raspredelenii, svyazannykh s vykhodom nediskretnogo sluchainogo bluzhdaniya iz intervala”, Predelnye teoremy teorii veroyatnostei i smezhnye voprosy, Tr. IM SO AN SSSR, 1, Nauka, Novosibirsk, 1982, 18–25 | MR

[4] Lotov V. I., Khodzhibaev V. R., “O veroyatnosti razoreniya”, Izv. AN UzSSR. Ser. fiz.-mat. nauk, 1980, no. 3, 28–34 | MR | Zbl

[5] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 1, Mir, M., 1984 | Zbl

[6] Emery D. J., “Exit problem for a spectrally positive processes”, Adv. in Appl. Probab., 5 (1973), 498–520 | DOI | MR | Zbl

[7] Kemperman J. H. B., “A Wiener–Hopf type method for a general random walk with a two-sided boundary”, Ann. Math. Statist., 34:4 (1963), 1169–1193 | DOI | MR