In this paper we analyze the so-called Parisian ruin probability that happens when the surplus process stays below zero longer than fixed amount of time \zeta>0. We focus on general spectrally negative L\'{e}vy insurance risk process. For this class of processes we identify the expression for the ruin probability in terms of some other quantities that could be possibly calculated explicitly in many models. We find its Cram\'{e}r-type and convolution-equivalent asymptotics when reserves tends to infinity. Finally, we analyze few explicit examples.
In this paper we analyze so-called Parisian ruin probability that happens when surplus process stays below zero longer than fixed amount of time \zeta>0. We focus on general spectrally negative L\'{e}vy insurance risk process. For this class of processes we identify expression for ruin probability in terms of some other quantities that could be possibly calculated explicitly in many models. We find its Cram\'{e}r-type and convolution-equivalent asymptotics when reserves tends to infinity. Finally, we analyze few explicit examples.
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