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Draw-down Parisian ruin for spectrally negative Lévy processes

Published online by Cambridge University Press:  03 December 2020

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

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.

Type
Original Article
Copyright
© Applied Probability Trust 2020

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