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2008 The Law of the Hitting Times to Points by a Stable Lévy Process with No Negative Jumps
Goran Peskir
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Electron. Commun. Probab. 13(none): 653-659 (2008). DOI: 10.1214/ECP.v13-1431

Abstract

Let $X=(X_t)_{t \ge 0}$ be a stable Lévy process of index $\alpha \in $ with the Lévy measure $\nu(dx) = (c/x^{1+\alpha}) I_{ dx$ 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: $$f_{\tau_x}(t) = \frac{x^{\alpha-1}}{\pi ( \Gamma(-\alpha) t)^{2-1/\alpha}} \sum_{n=1}^\infty \bigg[(-1)^{n-1} \sin(\pi/\alpha) \frac{\Gamma(n-1/\alpha)}{\Gamma(\alpha n-1)} \Big(\frac{x^\alpha}{c \Gamma(-\alpha)t} \Big)^{n-1} $$ $$- \sin\Big(\frac{n \pi}{\alpha}\Big) \frac{\Gamma(1+n/\alpha)}{n!} \Big(\frac{x^\alpha}{c \Gamma(-\alpha)t}\Big)^{(n+1)/\alpha-1} \bigg]$$ 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 $\sup_{0 \le s \le t} X_s$ that makes a Laplace inversion amenable. A simpler series representation for $f_{\tau_x}$ is also known to be valid when $x<0$.

Citation

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Goran Peskir. "The Law of the Hitting Times to Points by a Stable Lévy Process with No Negative Jumps." Electron. Commun. Probab. 13 653 - 659, 2008. https://doi.org/10.1214/ECP.v13-1431

Information

Accepted: 19 December 2008; Published: 2008
First available in Project Euclid: 6 June 2016

zbMATH: 1193.60066
MathSciNet: MR2466193
Digital Object Identifier: 10.1214/ECP.v13-1431

Subjects:
Primary: 45D05, 60G52
Secondary: 26A33, 45E99, 60J75

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