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Draw-down Parisian ruin for spectrally negative Lévy processes

Part of: Markov processes Stochastic processes Distribution theory - Probability

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: wwywang@xmu.edu.cn
**Postal address: Department of Mathematics and Statistics, Concordia University, Montreal, Canada. Email: xiaowen.zhou@concordia.ca
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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
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Copyright
© Applied Probability Trust 2020

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