Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-24T03:30:32.365Z Has data issue: false hasContentIssue false
  • English
  • Français

Draw-down Parisian ruin for spectrally negative Lévy processes

Part of: Markov processes Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  03 December 2020

Wenyuan Wang  and
Xiaowen Zhou
Wenyuan Wang*
Affiliation:
Xiamen University
Xiaowen Zhou*
Affiliation:
Concordia University
*
*Postal address: School of Mathematical Sciences, Xiamen University, Fujian361005, People’s Republic of China. Email: [email protected]
**Postal address: Department of Mathematics and Statistics, Concordia University, Montreal, Canada. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Draw-down time for a stochastic process is the first passage time of a draw-down level that depends on the previous maximum of the process. In this paper we study the draw-down-related Parisian ruin problem for spectrally negative Lévy risk processes. Intuitively, a draw-down Parisian ruin occurs when the surplus process has continuously stayed below the dynamic draw-down level for a fixed amount of time. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit problems via excursion theory. We also find an expression for the potential measure for the process killed at the draw-down Parisian time. As applications, we obtain new results for spectrally negative Lévy risk processes with dividend barrier and with Parisian ruin.

Keywords

Spectrally negative Lévy processreflected processParisian ruindraw-down timedraw-down Parisian ruinpotential measureexcursion theoryrisk process

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60E10: Characteristic functions; other transforms 60J35: Transition functions, generators and resolvents
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 4 , December 2020 , pp. 1164 - 1196
Check for updates
Copyright
© Applied Probability Trust 2020

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

Avram, F., Vu, N. and Zhou, X. (2017). On taxed spectrally negative Lévy processes with draw-down stopping. Insurance Math. Econom. 76, 69–74.10.1016/j.insmatheco.2017.06.005CrossRefGoogle Scholar
Baurdoux, E., Pardo, J., Perez, J. and Renaud, J. (2016). Gerber–Shiu functionals at Parisian ruin for Lévy insurance risk processes. J. Appl. Prob. 53, 572–584.10.1017/jpr.2016.21CrossRefGoogle Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Chan, T., Kyprianou, A. and Savov, M. (2011). Smoothness of scale functions for spectrally negative Lévy processes. Prob. Theory Relat. Fields 150, 691–708.10.1007/s00440-010-0289-4CrossRefGoogle Scholar
Chesney, M., Jeanblanc, M. and Yor, M. (1997). Brownian excursions and Parisian barrier options. Adv. Appl. Prob. 29, 165–184.10.2307/1427865CrossRefGoogle Scholar
Cheung, E. and Wong, J. (2017). On the dual risk model with Parisian implementation delays in dividend payments. Europ. J. Operat. Res. 257, 159–173.10.1016/j.ejor.2016.09.018CrossRefGoogle Scholar
Cohen, A. (2007). Numerical Methods for Laplace Transform Inversion. Springer, New York.Google Scholar
Czarna, I. (2016). Parisian ruin probability with a lower ultimate bankrupt barrier. Scand. Actuarial J. 2016, 319–337.10.1080/03461238.2014.926288CrossRefGoogle Scholar
Czarna, I. and Palmowski, Z. (2011). Ruin probability with Parisian delay for a spectrally negative Lévy risk process. J. Appl. Prob. 48, 984–1002.10.1017/S0021900200008573CrossRefGoogle Scholar
Czarna, I. and Palmowski, Z. (2014). Dividend problem with Parisian delay for a spectrally negative Lévy risk process. J. Optimization Theory Appl. 161, 239–256.10.1007/s10957-013-0283-yCrossRefGoogle Scholar
Czarna, I. and Renaud, J. (2016). A note on Parisian ruin with an ultimate bankruptcy level for Lévy insurance risk processes. Statis. Prob. Lett. 113, 54–61.10.1016/j.spl.2016.02.018CrossRefGoogle Scholar
Dassios, A. and Wu, S. (2009). Parisian ruin with exponential claims. Working paper, London School of Economics. Available at http://stats.lse.ac.uk/angelos.Google Scholar
Dassios, A. and Wu, S. (2009). Semi-Markov model for excursions and occupation time of Markov processes. Working paper, London School of Economics. Available at http://stats.lse.ac.uk/angelos.Google Scholar
Frostig, E. and Keren-Pinhasik, A. (2020). Parisian ruin with Erlang delay and a lower bankruptcy barrier. Methodology Comput. Appl. Prob. 22, 101–134.10.1007/s11009-019-09693-wCrossRefGoogle Scholar
Kuznetsov, A., Kyprianou, A. 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. 97–186.10.1007/978-3-642-31407-0_2CrossRefGoogle Scholar
Kyprianou, A. (2014). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.10.1007/978-3-642-37632-0CrossRefGoogle Scholar
Landriault, D., Renaud, J. and Zhou, X. (2014). Insurance risk models with Parisian implementation delays. Methodology Comput. Appl. Prob. 16, 583–607.10.1007/s11009-012-9317-4CrossRefGoogle Scholar
Lehoczky, J. P. (1977). Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Prob. 5, 601–607.10.1214/aop/1176995770CrossRefGoogle Scholar
Li, B., Vu, N. and Zhou, X. (2019). Exit problems for general draw-down times of spectrally negative Lévy processes. J. Appl. Prob. 56, 441–457.10.1017/jpr.2019.31CrossRefGoogle Scholar
Lkabous, M., Czarna, I. and Renaud, J. (2017). Parisian ruin for a refracted Lévy process. Insurance Math. Econom. 74, 153–163.10.1016/j.insmatheco.2017.03.005CrossRefGoogle Scholar
Loeffen, R., Czarna, I. and Palmowski, Z. (2013). Parisian ruin probability for spectrally negative Lévy processes. Bernoulli 19, 599–609.10.3150/11-BEJ404CrossRefGoogle Scholar
Loeffen, R., Palmowski, Z. and Surya, B. A. (2018). Discounted penalty function at Parisian ruin for Lévy insurance risk process. Insurance Math. Econom. 83, 190–197.10.1016/j.insmatheco.2017.10.008CrossRefGoogle Scholar
Moorthy, M. (1995). Numerical inversion of two-dimensional Laplace transforms—Fourier series representation. Appl. Numer. Math. 17, 119–127.10.1016/0168-9274(95)00015-MCrossRefGoogle Scholar
Pistorius, M. (2007). An excursion-theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. In Séminaire de Probabilités XL, Springer, Berlin, Heidelberg, pp. 287–307.10.1007/978-3-540-71189-6_15CrossRefGoogle Scholar
Renaud, J. and Zhou, X. (2007). Distribution of the present value of dividend payments in a Lévy risk model. J. Appl. Prob. 44, 420–427.10.1239/jap/1183667411CrossRefGoogle Scholar
Surya, B. A. (2019). Parisian excursion below a fixed level from the last record maximum of Lévy insurance risk process. In 2017 MATRIX Annals, Springer, Cham, pp. 311–326.10.1007/978-3-030-04161-8_21CrossRefGoogle Scholar
Wang, W. and Zhou, X. (2020). A draw-down reflected spectrally negative Lévy process. To appear in J. Theoret. Prob.Google Scholar
Zhou, X. (2007). Exit problems for spectrally negative Lévy processes reflected at either the supremum or the infimum. J. Appl. Prob. 44, 1012–1030.10.1239/jap/1197908821CrossRefGoogle Scholar