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Fluctuations of Omega-killed level-dependent spectrally negative Lévy processes

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  • Palmowski, Zbigniew
  • Şimşek, Meral
  • Papaioannou, Apostolos D.

Abstract

In this paper, we solve exit problems for a level-dependent Lévy process which is exponentially killed with a killing intensity that depends on the present state of the process. Moreover, we analyse the respective resolvents. All identities are given in terms of new generalisations of scale functions (counterparts of the scale function from the theory of Lévy processes), which are solutions of Volterra integral equations. Furthermore, we obtain similar results for the reflected level-dependent Lévy processes. The existence of the solution of the stochastic differential equation for reflected level-dependent Lévy processes is also discussed. Finally, to illustrate our result, the probability of bankruptcy is obtained for an insurance risk process.

Suggested Citation

  • Palmowski, Zbigniew & Şimşek, Meral & Papaioannou, Apostolos D., 2025. "Fluctuations of Omega-killed level-dependent spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 185(C).
    • Handle: RePEc:eee:spapps:v:185:y:2025:i:c:s0304414925000584
    DOI: 10.1016/j.spa.2025.104617
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    References listed on IDEAS

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    1. Czarna, Irmina & Pérez, José-Luis & Rolski, Tomasz & Yamazaki, Kazutoshi, 2019. "Fluctuation theory for level-dependent Lévy risk processes," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5406-5449.
    2. Pérez, José-Luis & Yamazaki, Kazutoshi, 2017. "Refraction–Reflection Strategies In The Dual Model," ASTIN Bulletin, Cambridge University Press, vol. 47(1), pages 199-238, January.
    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. Li, Bo & Palmowski, Zbigniew, 2018. "Fluctuations of Omega-killed spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 128(10), pages 3273-3299.
    5. M. R. Pistorius, 2004. "On Exit and Ergodicity of the Spectrally One-Sided Lévy Process Reflected at Its Infimum," Journal of Theoretical Probability, Springer, vol. 17(1), pages 183-220, January.
    6. Loeffen, Ronnie L. & Renaud, Jean-François & Zhou, Xiaowen, 2014. "Occupation times of intervals until first passage times for spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1408-1435.
    7. Landriault, David & Renaud, Jean-François & Zhou, Xiaowen, 2011. "Occupation times of spectrally negative Lévy processes with applications," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2629-2641, November.
    8. Hansjoerg Albrecher & Jevgenijs Ivanovs, 2013. "Power identities for L\'evy risk models under taxation and capital injections," Papers 1310.3052, arXiv.org, revised Mar 2014.
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