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 Lévy measure has density $c{x}^{-\alpha -1}$ on $(0,\infty )$, and $S_1={\sup }_{0<t\le 1}{X}_t$, it is known that $P(S_1>x)\backsim A{\alpha }^{-1}{x}^{-\alpha }$ as $x\to \infty$ and $P(S_1\le x)\backsim B{\alpha }^{-1}{\rho }^{-1}{x}^{\alpha \rho }$ as $x\downarrow 0$. [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 $m(x)\backsim A{x}^{-(\alpha +1)}$ as $x\to \infty$ and $m(x)\backsim B{x}^{\alpha \rho -1}$ as $x\downarrow 0$. Similar results are obtained for related densities.

Published in at http://dx.doi.org/10.1214/09-AOP479 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• The law of the supremum of a stable Lévy process with no negative jumps

Cited by in corpus (2)

• $\epsilon$-strong simulation of the convex minorants of stable processes and meanders
• Joint density of a stable process and its supremum: regularity and upper bounds