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The Law of the Hitting Times to Points by a Stable Lévy Process with No Negative Jumps
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Goran Peskir, The University of Manchester
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Abstract
Let $X=(X_t)_{t ge 0}$ be a stable L'evy process of index $alpha
in (1,2)$ with the L'evy measure $nu(dx) = (c/x^{1+alpha}):!
I_{(0,infty)}(x), dx$ for $c>0$, let $x>0$ be given and fixed, and
let $tau_x = inf, {, t>0 : X_t=x, }$ denote the first hitting
time of $X$ to $x$. Then the density function $f_{tau_x}$ of
$tau_x$ admits the following series representation:
begin{align} notag
f_{tau_x}(t) = frac{x^{alpha-1}}{pi:! (c;! Gamma(-alpha)
;!t)^{2-1/alpha}} sum_{n=1}^infty bigg[:! &(-1)^{n-1}
sin(pi/alpha), frac{Gamma(n!-!1/alpha)}{Gamma(alpha
n!-!1)}, Big(frac{x^alpha}{c;! Gamma(-alpha);!t}
Big)^{!n-1} notag &- sinBig(frac{n pi}{alpha}Big),
frac{Gamma(1!+!n/alpha)}{n!}, Big(frac{x^alpha}{c;!
Gamma(-alpha);!t}Big)^{!(n+1)/alpha-1}, bigg]
end{align}
for $t>0$. In particular, this yields $f_{tau_x}(0+)=0$ and
begin{equation} notag
f_{tau_x}(t) sim frac{x^{alpha-1}}{Gamma(alpha!-!1),
Gamma(1/alpha)}, (c;! Gamma(-alpha);!t)^{-2+1/alpha}
end{equation}
as $t rightarrow infty$. The method of proof exploits a simple
identity linking the law of $tau_x$ 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 $f_{tau_x}$ is also
known to be valid when $x<0$.
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Full text: PDF
Pages: 653-659
Published on: December 19, 2008
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