A note on the supremum of a stable process

TL;DR

This paper investigates the asymptotic behavior of the density of the supremum of a spectrally positive stable process, showing it decays as a power law with exponent (+) at infinity.

Contribution

It establishes the asymptotic decay rate of the supremum's density for stable processes, extending known tail probability results to the density.

Findings

Density of supremum behaves as c x^{-(+)} for large x

Confirms the supremum's density is continuous and has a specific tail decay

Provides a precise asymptotic for the density at infinity

Abstract

If $X$ is a spectrally positive stable process of index $\alpha\in(1,2)$ whose L\'{e}vy measure has density $cx^{-\alpha-1}$ on $(0,\infty),$ and $S_1=\sup_{0<t\leq1}X_t,$ it is known that $P(S_1>x)\backsim c\alpha^{-1}x^{-\alpha}$ as $x\to\infty.$ It is also known that $S_1$has a continuous density, $s$ say. The point of this note is to show that $s(x)\backsim cx^{-(\alpha+1)}$ as $x\to\infty.$