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On doubly reflected completely asymmetric Lévy processes

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  • Pistorius, M. R.

Abstract

Consider a completely asymmetric Lévy process X and let Z be X reflected at 0 and at a>0. In applied probability (e.g. The Single Server Queue, 2nd Edition, North-Holland, Amsterdam, 1982) the process Z turns up in the study of the virtual waiting time in an M/G/1-queue with finite buffer a or the water level in a finite dam of size a. We find an expression for the resolvent density of Z. We show Z is positive recurrent and determine the invariant measure. Using the regenerative property of Z, we determine the asymptotic law of for an appropriate class of functions f. Finally, the long time average of the local time of Z in x[set membership, variant][0,a] is studied.

Suggested Citation

  • Pistorius, M. R., 2003. "On doubly reflected completely asymmetric Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 131-143, September.
    • Handle: RePEc:eee:spapps:v:107:y:2003:i:1:p:131-143
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    References listed on IDEAS

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    1. Avram, Florin & Chan, Terence & Usabel, Miguel, 0. "On the valuation of constant barrier options under spectrally one-sided exponential Lévy models and Carr's approximation for American puts," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 75-107, July.
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    Cited by:

    1. Baurdoux, Erik J. & Yamazaki, Kazutoshi, 2015. "Optimality of doubly reflected Lévy processes in singular control," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2727-2751.
    2. Florin Avram & Zbigniew Palmowski & Martijn R. Pistorius, 2007. "On the optimal dividend problem for a spectrally negative L\'{e}vy process," Papers math/0702893, arXiv.org.
    3. Pérez, José-Luis & Yamazaki, Kazutoshi, 2018. "On the refracted–reflected spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 306-331.
    4. Bo, Lijun & Song, Renming & Tang, Dan & Wang, Yongjin & Yang, Xuewei, 2012. "Lévy risk model with two-sided jumps and a barrier dividend strategy," Insurance: Mathematics and Economics, Elsevier, vol. 50(2), pages 280-291.
    5. Czarna, Irmina & Pérez, José-Luis & Yamazaki, Kazutoshi, 2018. "Optimality of multi-refraction control strategies in the dual model," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 148-160.

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