The law of the supremum of a stable L\'{e}vy process with no negative jumps

TL;DR

This paper characterizes the density of the supremum of a stable Lévy process with no negative jumps using integral and fractional differential equations, providing explicit series representations for the density functions.

Contribution

It introduces a novel characterization of the supremum's density via Volterra integral and fractional differential equations, leading to explicit series solutions.

Findings

Explicit series representation for the density of the supremum

Unique solution to integral and fractional differential equations

Connection between supremum density and first entry time density

Abstract

Let $X=(X_t)_{t\ge0}$ be a stable L\'{e}vy process of index $\alpha \in(1,2)$ 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 $\tau_x$ of $X$ into $[x,\infty)$, this further translates into an explicit series representation for the density function of $\tau_x$.