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

Abstract

If $X$ is a stable process of index $α\in(0,2)$ whose Lévy measure has density $cx^{-α-1}$ on $(0,\infty)$, and $S_1=\sup_{0 x)\backsim Aα^{-1}x^{-α}$ as $x\to\infty$ and $P(S_1\leq x)\backsim Bα^{-1}ρ^{-1}x^{αρ}$ as $x\downarrow0$. [Here $ρ=P(X_1>0)$ 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 $m(x)\backsim Ax^{-(α+1)}$ as $x\to\infty$ and $m(x)\backsim Bx^{αρ-1}$ as $x\downarrow0$. Similar results are obtained for related densities.