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We study a first passage time of a L\'evy process over a positive constantlevel. In the spectrally negative case we give conditions for absolutelycontinuity of the distributions of the first passage times. The tailasymptotics of their densities are also clarified, where the asymptotics dependon tail behaviour of the corresponding L\'evy measures. We apply our results tothe mathematical finance, in particular, the credit default swap pricing.

Asymptotics of densities of first passage times for spectrally negative L\'evy processes