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On the Parisian ruin of the dual Lévy risk model. (English) Zbl 1416.91226

Summary: In this paper, we investigate the Parisian ruin problem of the general dual Lévy risk model. Unlike the usual concept of ultimate ruin, allowing the surplus level to be negative within a prespecified period indicates that the deficit at Parisian ruin is not necessarily equal to zero. Hence, we consider a Gerber-Shiu type expected discounted penalty function at the Parisian ruin and obtain an explicit expression for this function under the dual Lévy risk model. As particular cases, we calculate the Parisian ruin probability and the expected discounted \(k\)th moments of the deficit at the Parisian ruin for the compound Poisson dual risk model and a drift-diffusion model. Numerical examples are given to illustrate the behavior of Parisian ruin and the expected discounted deficit at Parisian ruin.

MSC:

91B30 Risk theory, insurance (MSC2010)
60G51 Processes with independent increments; Lévy processes
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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