Parisian Ruin with Erlang Delay and a Lower Bankruptcy Barrier

Abstract

Parisian ruin occurs once the surplus stays continuously below zero for a given period. We consider the spectrally negative Levy risk process where ruin is declared either at the first time that the reserve stays continuously below zero for an exponentially or mixed Erlang distributed random variable, or once it reaches a given negative threshold. We consider the Laplace transform of the time to ruin and the Laplace transform of the time that the process is negative.