Skip to main content
Log in

On the distribution of the time of the first exit from an interval and the value of a jump over the boundary for processes with independent increments and random walks

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

For a homogeneous process with independent increments, we determine the integral transforms of the joint distribution of the first-exit time from an interval and the value of a jump of a process over the boundary at exit time and the joint distribution of the supremum, infimum, and value of the process.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. V. Skorokhod, Random Processes with Independent Increments [in Russian], Nauka, Moscow (1964).

    Google Scholar 

  2. I. I. Gikhman and A. V. Skorokhod, Theory of Random Processes [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  3. E. A. Pecherskii and B. A. Rogozin, “On joint distributions of random variables related to fluctuations of a process with independent increments,” Teor. Ver. Primen., 14, No. 3, 431–444 (1964).

    Google Scholar 

  4. A. A. Borovkov, Probability Processes in Queuing Theory [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  5. I. G. Petrovskii, Lectures on the Theory of Integral Equations [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  6. Yu. V. Borovskikh, “Complete asymptotic decompositions for the resolvent of a semicontinuous process with independent increments with absorption and distributions of ruin probability,” in: Asymptotic Methods in Probability Theory [in Russian], Kiev (1979), pp. 10–21.

  7. D. J. Emery, “Exit problem for a spectrally positive process,” Adv. Appl. Probab., 5, 498–520 (1973).

    MATH  MathSciNet  Google Scholar 

  8. E. A. Pecherskii, “Some identities related to the exit of a random walk from a segment or from a half-interval,” Teor. Ver. Primen., 19, No. 1, 104–119 (1974).

    MATH  Google Scholar 

  9. V. N. Suprun and V. M. Shurenkov, “On the resolvent of a process with independent increments that terminates at the moment when it hits the negative semiaxis,” in: Investigations in the Theory of Random Processes [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1975), pp. 170–174.

    Google Scholar 

  10. V. N. Suprun, “Ruin problem and the resolvent of a terminating process with independent increments,” Ukr. Mat. Zh., 28, No. 1, 53–61 (1976).

    MATH  MathSciNet  Google Scholar 

  11. V. S. Korolyuk, Boundary-Value Problems for Complicated Poisson Processes [in Russian], Naukova Dumka, Kiev (1975).

    Google Scholar 

  12. V. F. Kadankov and T. V. Kadankova, “On the distribution of duration of stay in an interval of the semicontinuous process with independent increments,” Random Oper. Stochast. Equat., 12, No. 4, 365–388 (2004).

    MathSciNet  Google Scholar 

  13. T. V. Kadankova, “On the distribution of the number of the intersections of a fixed interval by the semi-continuous process with independent increments,” Theor. Stochast. Proc., No. 1–2, 73–81 (2003).

  14. T. V. Kadankova, “On the joint distribution of the supremum, infimum, and value of a semicontinuous process with independent increments,” Teor. Imov. Mat. Statist., Issue 70, 56–65 (2004).

    Google Scholar 

  15. K. Itô and H. P. McKean, Diffusion Processes and Their Sample Paths, Springer, Berlin (1965).

    Google Scholar 

  16. G. Doetsch, Anleitung zum Praktischen Gebrauch der Laplace-Transformation, Oldenbourg, München (1956).

  17. P. Lévy, Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris (1948).

    Google Scholar 

  18. F. Spitzer, Principles of Random Walk, van Nostrand, Princeton (1964).

    Google Scholar 

  19. T. V. Kadankova, “Two-boundary problems for a random walk with geometrically distributed negative jumps,” Teor. Imov. Mat. Statist., Issue 68, 60–71 (2003).

  20. L. Takács, Combinatorial Methods in the Theory of Stochastic Processes, Wiley, New York (1967).

    Google Scholar 

  21. A. de Moivre, “De mensura sortis, seu, de probabilitate eventuum in ludis a casu fortuito pendentibus,” Philos. Trans. London, 27, 213–264 (1711).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

__________

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1359–1384, October, 2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kadankov, V.F., Kadankova, T.V. On the distribution of the time of the first exit from an interval and the value of a jump over the boundary for processes with independent increments and random walks. Ukr Math J 57, 1590–1620 (2005). https://doi.org/10.1007/s11253-006-0016-6

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-006-0016-6

Keywords

Navigation