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Ruin Probability with Parisian Delay for a Spectrally Negative Lévy Risk Process

Part of: Stochastic processes Markov processes Stochastic systems and control

Published online by Cambridge University Press:  14 July 2016

Irmina Czarna  and
Zbigniew Palmowski
Irmina Czarna*
Affiliation:
University of Wrocław
Zbigniew Palmowski*
Affiliation:
University of Wrocław
*
∗ Postal address: Department of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
∗ Postal address: Department of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
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Abstract

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In this paper we analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ > 0. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we derive an expression for the ruin probability in terms of quantities that can be calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tend to ∞. Finally, we analyze some explicit examples.

Keywords

Lévy processruin probabilityasymptoticsParisian ruinrisk process

MSC classification

Secondary: 60J99: None of the above, but in this section 93E20: Optimal stochastic control 60G51: Processes with independent increments; L'evy processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 984 - 1002
Copyright
Copyright © Applied Probability Trust 2011 

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