Explicit formula for the supremum distribution of a spectrally negative stable process

Explicit formula for the supremum distribution of a spectrally negative stable process

In this article we get simple formulas for IE sups≤tX(s) where X is a spectrally positive or negative Lévy process with infinite variation. As a consequence we derive a generalization of the well-known formula for the supremum distribution of Wiener process that is we obtain IP(sups≤t Zα(s) ≥ u) = α IP(Zα(t) ≥ u) for u ≥ 0 where Zα is a spectrally negative α-stable Lévy process with 1 < α ≤ 2 w...

The distribution of the supremum for spectrally asymmetric Lévy processes

In this article we derive formulas for the probability IP(supt≤T X(t) > u), T > 0 and IP(supt<∞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 inft≤T Y (t) and Y (T ) where Y is ...

A Note on the Supremum of a Stable Process

A Convergent Series Representation for the Density of the Supremum of a Stable Process

We study the density of the supremum of a strictly stable Lévy process. We prove that for almost all values of the index α – except for a dense set of Lebesgue measure zero – the asymptotic series which were obtained in [13] are in fact absolutely convergent series representations for the density of the supremum.

A sample size formula for the supremum log-rank statistic.

An advantage of the supremum log-rank over the standard log-rank statistic is an increased sensitivity to a wider variety of stochastic ordering alternatives. In this article, we develop a formula for sample size computation for studies utilizing the supremum log-rank statistic. The idea is to base power on the proportional hazards alternative, so that the supremum log rank will have the same p...