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The law of the hitting times to points by a stable Lévy process with no negative jumps. (English) Zbl 1193.60066

Summary: Let \(X=(X_t)_{t \geq 0}\) be a stable Lévy process of index \(alpha\in (1,2)\) with the Lévy 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:
\[ \begin{split} f_{\tau_x}(t) = \frac{x^{\alpha-1}}{\pi(c\Gamma(-\alpha)t)^{2-1/\alpha}} \sum_{n=1}^\infty \left[(-1)^{n-1} \sin(\pi/\alpha), \frac{\Gamma(n1/\alpha)}{\Gamma(\alpha n-1)}, \left(\frac{x^\alpha}{c\Gamma(-\alpha)t}\right)^{n-1}\right.\\ \left.-\sin\left(\frac{n \pi}{\alpha}\right), \frac{\Gamma(1+n/\alpha)}{n!}\left(\frac{x^\alpha}{c\Gamma(-\alpha)t}\right)^{(n+1)/\alpha-1}\right]\end{split} \]
for \(t>0\). In particular, this yields \(f_{\tau_x}(0+)=0\) and
\[ f_{\tau_x}(t) \sim \frac{x^{\alpha-1}}{\Gamma(\alpha-1)\Gamma(1/\alpha)}\;(c\Gamma(-\alpha)t)^{-2+1/\alpha} \]
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 \(\sup_{0\leq s\leq t} X_s\) that makes a Laplace inversion amenable. A simpler series representation for \(f_{\tau_x}\) is also known to be valid when \(x<0\).

MSC:

60G52 Stable stochastic processes
60G51 Processes with independent increments; Lévy processes
60J75 Jump processes (MSC2010)
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