Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T03:01:27.805Z Has data issue: false hasContentIssue false
  • English
  • Français

On the last exit times for spectrally negative Lévy processes

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  22 June 2017

Yingqiu Li ,
Chuancun Yin  and
Xiaowen Zhou
Yingqiu Li*
Affiliation:
Changsha University of Science and Technology
Chuancun Yin*
Affiliation:
Qufu Normal University
Xiaowen Zhou*
Affiliation:
Concordia University
*
* Postal address: School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114, China.
** Postal address: School of Statistics, Qufu Normal University, Qufu, Shandong 273165, China.
*** Postal address: Department of Mathematics and Statistics, Concordia University, Montreal, Quebec H3G 1M8, Canada. Email address: xiaowen.zhou@concordia.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Using a new approach, for spectrally negative Lévy processes we find joint Laplace transforms involving the last exit time (from a semiinfinite interval), the value of the process at the last exit time, and the associated occupation time, which generalize some previous results.

Keywords

Spectrally negative Lévy processlast exit timeGerber–Shiu functionoccupation time

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 474 - 489
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Albrecher, H., Ivanovs, J. and Zhou, X. (2016). Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli 22, 1364–1382. CrossRefGoogle Scholar
[2] Baurdoux, E. J. (2009). Last exit before an exponential time for spectrally negative Lévy processes. J. Appl. Prob. 46, 542–558. CrossRefGoogle Scholar
[3] Chiu, S. N. and Yin, C. (2005). Passage times for a spectrally negative Lévy process with applications to risk theory. Bernoulli 11, 511–522. Google Scholar
[4] Gerber, H. U. (1990). When does the surplus reach a given target? Insurance Math. Econom. 9, 115–119. Google Scholar
[5] Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2012). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II (Lecture Notes Math. 2061), Springer, Heidelberg, pp. 97–186. Google Scholar
[6] Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Berlin. CrossRefGoogle Scholar
[7] Li, B. and Cai, C. (2016). Occupation times of intervals until last passage times for spectrally negative Lévy processes. Preprint. Available at https://arxiv.org/abs/1605.07709v2. Google Scholar
[8] Li, B. and Zhou, X. (2013). The joint Laplace transforms for diffusion occupation times. Adv. Appl. Prob. 45, 1049–1067. Google Scholar
[9] Li, Y. and Zhou, X. (2014). On pre-exit joint occupation times for spectrally negative Lévy processes. Statist. Prob. Lett. 94, 48–55. Google Scholar
[10] Loeffen, R., Renaud, J.-F. and Zhou, X. (2014). Occupation times of intervals until first passage times for spectrally negative Lévy processes. Stoch. Process. Appl. 124, 1408–1435. CrossRefGoogle Scholar
[11] Pérez, J.-L. and Yamazaki, K. (2016). On the refracted-reflected spectrally negative Lévy processes. Preprint. Available at https://arxiv.org/abs/1511.06027v1. Google Scholar
[12] Zhang, H. (2015). Occupation times, drawdowns, and drawups for one-dimensional regular diffusions. Adv. Appl. Prob. 47, 210–230. CrossRefGoogle Scholar
[13] Zhang, H. and Hadjiliadis, O. (2012). Drawdowns and the speed of market crash. Methodology Comput. Appl. Prob. 14, 739–752. CrossRefGoogle Scholar
[14] Zhou, X. (2007). Exit problems for spectrally negative Lévy processes reflected at either the supremum or the infimum. J. Appl. Prob. 44, 1012–1030. Google Scholar