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Exit problems for general draw-down times of spectrally negative Lévy processes

Part of: Stochastic processes Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  30 July 2019

Bo Li ,
Nhat Linh Vu  and
Xiaowen Zhou
Bo Li*
Affiliation:
Nankai University
Nhat Linh Vu*
Affiliation:
Concordia University
Xiaowen Zhou*
Affiliation:
Concordia University
*
*Postal address: School of Mathematics and LPMC, Nankai University, Tianjin 300071, PR China.
**Postal address: Department of Mathematics and Statistics, Concordia University, Montreal, Canada.
**Postal address: Department of Mathematics and Statistics, Concordia University, Montreal, Canada.
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Abstract

For spectrally negative Lévy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functions for the spectrally negative Lévy processes.

Keywords

Spectrally negative Lévy processdraw-down timeexit problempotential measurecreeping timehitting time

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60E10: Characteristic functions; other transforms 60J35: Transition functions, generators and resolvents
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 2 , June 2019 , pp. 441 - 457
Copyright
© Applied Probability Trust 2019 

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References

Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215–238.Google Scholar
Avram, F., Vu, N.L. and Zhou, X. (2017). On taxed spectrally negative Lévy processes with draw-down stopping. Insurance Math. Econom. 76, 69–74.CrossRefGoogle Scholar
Azéma, J. and Yor, M. (1979). Une solution simple au problème de Skorokhod. In Séminaire de Probabilités XIII (Lecture Notes Math. 721), pp. 90–115. Springer, Berlin and Heidelberg.CrossRefGoogle Scholar
Bertoin, J. (1996). Lévy Processes, Cambridge Tracts in Mathematics. Cambridge University Press.Google Scholar
Carraro, L., Karoui, N. E. and ObłόJ, J. (2012). On Azéma–Yor processes, their optimal properties and the Bachelier–drawdown equation. Ann. Prob. 40, 372–400.CrossRefGoogle Scholar
Chan, T., Kyprianou, A. E. and Savov, M. (2011). Smoothness of scale functions for spectrally negative Lévy processes. Prob. Theory Rel. Fields 150, 691–708.CrossRefGoogle Scholar
Ivanovs, J. and Palmowski, Z. (2012). Occupation densities in solving exit problems for Markov additive processes and their reflections. Stoch. Process. Appl. 122, 3342–3360.CrossRefGoogle Scholar
Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications. Springer, Berlin and Heidelberg.CrossRefGoogle Scholar
Landriault, D., Li, B. and Li, S. (2017). Drawdown analysis for the renewal insurance risk process. Scand. Actuar. J. 2017, 267–285.Google Scholar
Landriault, D., Li, B. and Zhang, H. (2017). On magnitude, asymptotics and duration of drawdowns for Lévy models. Bernoulli 23, 432–458.CrossRefGoogle Scholar
Lehoczky, J. P. (1977). Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Prob. 5, 601–607.CrossRefGoogle Scholar
Mijatović, A. and Pistorius, M. R. (2012). On the drawdown of completely asymmetric Lévy processes. Stoch. Process. Appl. 122, 3812–3836.CrossRefGoogle Scholar
Pistorius, M. R. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Probab. 17, 183–220.CrossRefGoogle Scholar
Pistorius, M. R. (2012). An excursion-theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. In Séminaire de Probabilités XL. (Lecture Notes Math. 1899), pp. 287–307. Springer, Berlin and Heidelberg.CrossRefGoogle Scholar
Zhang, H. (2015). Occupation times, drawdowns, and drawups for one-dimensional regular diffusions. Adv. Appl. Prob. 47, 210–230.CrossRefGoogle 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.CrossRefGoogle Scholar