Parisian Excursion Below a Fixed Level from the Last Record Maximum of Lévy Insurance Risk Process
This paper presents some new results on Parisian ruin under Lévy insurance risk process, where ruin occurs when the process has gone below a fixed level from the last record maximum, also known as the high-water mark or drawdown, for a fixed consecutive periods of time. The law of ruin-time and the position at ruin is given in terms of their joint Laplace transforms. Identities are presented semi-explicitly in terms of the scale function and the law of the Lévy process. They are established using recent developments on fluctuation theory of drawdown of spectrally negative Lévy process. In contrast to the Parisian ruin of Lévy process below a fixed level, ruin under drawdown occurs in finite time with probability one.
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The author would like to thank a number of anonymous referees and associate editors for their useful suggestions and comments that improved the presentation of this paper. This paper was completed during the time the author visited the Hugo Steinhaus Center of Mathematics at Wrocław University of Science and Technology in Poland. The author acknowledges the support and hospitality provided by the Center. He thanks to Professor Zbigniew Palmowski for the invitation, and for some suggestions over the work discussed during the MATRIX Mathematics of Risk Workshop in Melbourne organized by Professors Konstantin Borovkov, Alexander Novikov and Kais Hamza to whom the author also like to thanks for the invitation. This research is financially supported by Victoria University PBRF Research Grants # 212885 and # 214168 for which the author is grateful.
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