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Advances in Applied Probability Advances in Applied Probability

Article contents

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
Corresponding
E-mail address:
wwywang@xmu.edu.cn
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 process reflected process Parisian ruin draw-down time draw-down Parisian ruin potential measure excursion theory risk 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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