The asymptotic behavior of densities related to the supremum of a stable process

TL;DR

This paper investigates the asymptotic behavior of the density of the supremum of a stable process, revealing how it behaves at zero and infinity, extending known tail probability results to density asymptotics.

Contribution

It establishes the asymptotic behavior of the supremum's density at zero and infinity, providing new insights beyond existing tail probability asymptotics for stable processes.

Findings

Density behaves like $Ax^{-( extalpha+1)}$ as $x o\infty$

Density behaves like $Bx^{ extalpha ho-1}$ as $x o 0$

Results extend tail probability asymptotics to density asymptotics.

Abstract

If $X$ is a stable process of index $\alpha\in(0,2)$ 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 $P(S_1>x)\backsim A\alpha ^{-1}x^{-\alpha}$ as $x\to\infty$ and $P(S_1\leq x)\backsim B\alpha^{-1}\rho^{-1}x^{\alpha\rho}$ as $x\downarrow0$. [Here $\rho =P(X_1>0)$ and $A$ and $B$ are known constants.] It is also known that $S_1$ has a continuous density, $m$ say. The main point of this note is to show that $m(x)\backsim Ax^{-(\alpha+1)}$ as $x\to\infty$ and $m(x)\backsim Bx^{\alpha\rho-1}$ as $x\downarrow0$. Similar results are obtained for related densities.