Skip to main content
SpringerLink
Log in

On Several Two-Boundary Problems for a Particular Class of Lévy Processes

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

Several two-boundary problems are solved for a special Lévy process: the Poisson process with an exponential component. The jumps of this process are controlled by a homogeneous Poisson process, the positive jump size distribution is arbitrary, while the distribution of the negative jumps is exponential. Closed form expressions are obtained for the integral transforms of the joint distribution of the first exit time from an interval and the value of the overshoot through boundaries at the first exit time. Also the joint distribution of the first entry time into the interval and the value of the process at this time instant are determined in terms of integral transforms.

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

Access this article

Log in via an institution

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. Bertoin, J.: Lévy Processes. Cambridge University Press (1996)

  2. Bertoin, J.: Exponential decay and ergodicity of completely asymmetric Lévy process in a finite interval. Ann. Appl. Probab. 7, 156–169 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Borovkov, A.A.: Stochastic Processes in Queueing Theory. Springer, Berlin (1976)

    MATH  Google Scholar 

  4. Bratiychuk, N.S., Gusak, D.V.: Boundary Problems for Processes with Independent Increments. Naukova Dumka, Kiev (1990) (in Russian)

  5. Ditkin, V.A., Prudnikov, A.P.: Operational Calculus. Moscow (1966). Russian edition

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Gihman, I.I., Skorokhod, A.V.: Theory of Stochastic Processes, vol. 2. Springer, Berlin (1975). Translated from the Russian by S. Kotz

    MATH  Google Scholar 

  8. Kadankov, V.F., Kadankova, T.V.: On the distribution of duration of stay in an interval of the semi-continuous process with independent increments. Random Oper. Stoch. Equ. (ROSE) 12(4), 365–388 (2004)

    MathSciNet  Google Scholar 

  9. Kadankov, V.F., Kadankova, T.V.: On the distribution of the first exit time from an interval and the value of overshoot through the boundaries for processes with independent increments and random walks. Ukr. Math. J. 10(57), 1359–1384 (2005)

    MathSciNet  Google Scholar 

  10. Kadankova, T.V.: On the distribution of the number of the intersections of a fixed interval by the semi-continuous process with independent increments. Theor. Stoch. Process. 1–2, 73–81 (2003)

    MathSciNet  Google Scholar 

  11. Kadankova, T.V.: On the joint distribution of supremum, infimum and the magnitude of a process with independent increments. Theor. Probab. Math. Stat. 70, 54–62 (2004)

    Google Scholar 

  12. Kyprianou, A.E.: A martingale review of some fluctuation theory for spectrally negative Lévy processes. Research report, Utrecht University (2003)

  13. Pecherskii, E.A.: Rogozin, B.A. (1969). On joint distributions of random variables associated with fluctuations of a process with independent increments. Theor. Probab. Appl. 14, 410–423

    Google Scholar 

  14. Petrovskii, I.G.: Lectures on the Theory of Integral Equations. Nauka, Moscow (1965). Russian edition

    Google Scholar 

  15. Pistorius, M.R.: A potential theoretical review of some exit problems of spectrally negative Lévy processes. Séminaire Probab. 38, 30–41 (2004)

    Google Scholar 

  16. Pistorius, M.R.: On exit and ergodicity of the spectrally negative Lévy process reflected in its infimum. J. Theor. Probab. 17, 183–220 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rogozin, B.A.: On distributions of functionals related to boundary problems for processes with independent increments. Theor. Probab. Appl. 11(4), 656–670 (1966)

    Article  MathSciNet  Google Scholar 

  18. Suprun, V.N.: Ruin problem and the resolvent of a terminating process with independent increments. Ukr. Math. J. 28(1), 53–61 (1976) (English transl.)

    Article  MATH  MathSciNet  Google Scholar 

  19. Suprun, V.N., Shurenkov, V.M.: On the resolvent of a process with independent increments terminating at the moment when it hits the negative real semiaxis. In: Studies in the Theory of Stochastic Processes, pp. 170–174. Institute of Mathematics, Academy of Sciences of UKrSSR, Kiev (1976)

    Google Scholar 

  20. Zolotarev, V.M.: The first passage time of a level and the behaviour at infinity for a class of processes with independent increments. Theor. Probab. Appl. 9(4), 653–664 (1964)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Statistics, Hasselt University, Agoralaan, Building D, 3590, Diepenbeek, Belgium

    T. Kadankova & N. Veraverbeke

Authors
  1. T. Kadankova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. Veraverbeke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Kadankova.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kadankova, T., Veraverbeke, N. On Several Two-Boundary Problems for a Particular Class of Lévy Processes. J Theor Probab 20, 1073–1085 (2007). https://doi.org/10.1007/s10959-007-0088-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-007-0088-8

Keywords

Search

Navigation