The distribution of the supremum for spectrally asymmetric lévy processes

Abstract

In this article we derive formulas for the probability IP(sup t≤T X(t) > u), T > 0 and IP(sup t X(t) > u) where X is a spectrally positive Lévy process with infinite variation. The formulas are generalizations of the well-known Takács formulas for stochastic processes with non-negative and interchangeable increments. Moreover, we find the joint distribution of t≤T Y (t) and Y (T) where Y is a spectrally negative Lévy process.