ON MOMENTS OF DOWNWARD PASSAGE TIMES FOR SPECTRALLY NEGATIVE LEVY PROCESSES

The existence of moments of first downward passage times of a spectrally negative Levy process is governed by the general dynamics of the Levy process, i.e. whether it is drifting to +infinity, -infinity, or oscillating. Whenever the Levy process drifts to +infinity, we prove that the Kth moment of the first passage time (conditioned to be finite) exists if and only if the (K + 1)th moment of the Levy jump measure exists. This generalizes a result shown earlier by Delbaen for Cramer-Lundberg risk processes. Whenever the Levy process drifts to -infinity, we prove that all moments of the first passage time exist, while for an oscillating Levy process we derive conditions for non-existence of the moments, and in particular we show that no integer moments exist.