The law of the supremum of a stable Lévy process with no negative jumps

Abstract

Let $X=(X_t)_{t\ge0}$ be a stable Lévy process of index $α\in(1,2)$ with no negative jumps and let $S_t=\sup_{0\le s\le t}X_s$ denote its running supremum for $t>0$. We show that the density function $f_t$ of $S_t$ can be characterized as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first-order Riemann--Liouville fractional differential equation satisfying a boundary condition at zero. This yields an explicit series representation for $f_t$. Recalling the familiar relation between $S_t$ and the first entry time $τ_x$ of $X$ into $[x,\infty)$, this further translates into an explicit series representation for the density function of $τ_x$.

Published in at http://dx.doi.org/10.1214/07-AOP376 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)