Discounted penalty function at Parisian ruin for Lévy insurance risk process

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Insurance: Mathematics and Economics, ISSN: 0167-6687, Vol: 83, Page: 190-197

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R. Loeffen; Z. Palmowski; B. A. Surya
Elsevier BV
Mathematics; Economics, Econometrics and Finance; Decision Sciences
article description
In the setting of a Lévy insurance risk process, we present some results regarding the Parisian ruin problem which concerns the occurrence of an excursion below zero of duration bigger than a given threshold r. First, we give the joint Laplace transform of ruin-time and ruin-position (possibly killed at the first-passage time above a fixed level b ), which generalizes known results concerning Parisian ruin. This identity can be used to compute the expected discounted penalty function via Laplace inversion. Second, we obtain the q -potential measure of the process killed at Parisian ruin. The results have semi-explicit expressions in terms of the q -scale function and the distribution of the Lévy process.