The distribution of the supremum for spectrally asymmetric Lévy processes

Abstract

In this article we derive formulas for the probability $P({\sup }_{t\le T}X(t)>u)$ $T>0$ and $P({\sup }_{t<\infty }X(t)>u)$ 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 ${\inf }_{t\le T}Y(t)$ and $Y(T)$ where $Y$ is a spectrally negative Lévy process.

References in corpus (1)

• Wiener-Hopf factorization and distribution of extrema for a family of Lévy processes