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

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.