The Law of the Hitting Times to Points by a Stable Lévy Process with No Negative Jumps

摘要

Let $X=(X_t)_{t ge 0}$ be a stable Levy process of index $lpha in (1,2)$ with the Levy measure $u(dx) = (c/x^{1+lpha}) I_{(0,infty)}(x) dx$ for $c>0$, let $x>0$ be given and fixed, and let $au_x = inf{ t>0 : X_t=x }$ denote the first hitting time of $X$ to $x$. Then the density function $f_{au_x}$ of $au_x$ admits the following series representation: $$f_{au_x}(t) = rac{x^{lpha-1}}{pi ( Gamma(-lpha) t)^{2-1/lpha}} sum_{n=1}^infty igg[(-1)^{n-1} sin(pi/lpha) rac{Gamma(n-1/lpha)}{Gamma(lpha n-1)} Big(rac{x^lpha}{c Gamma(-lpha)t} Big)^{n-1} $$ $$- sinBig(rac{n pi}{lpha}Big) rac{Gamma(1+n/lpha)}{n!} Big(rac{x^lpha}{c Gamma(-lpha)t}Big)^{(n+1)/lpha-1} igg]$$ for $t>0$. In particular, this yields $f_{au_x}(0+)=0$ and $$ f_{au_x}(t) sim rac{x^{lpha-1}}{Gamma(lpha-1), Gamma(1/lpha)} (c Gamma(-lpha)t)^{-2+1/lpha} $$ as $t ightarrow infty$. The method of proof exploits a simple identity linking the law of $au_x$ to the laws of $X_t$ and $sup_{0 le s le t} X_s$ that makes a Laplace inversion amenable. A simpler series representation for $f_{au_x}$ is also known to be valid when $x<0$.