The Law of the Hitting Times to Points by a Stable Lévy Process with No Negative Jumps
| Dublin Core | PKP Metadata Items | Metadata for this Document | |
| 1. | Title | Title of document | The Law of the Hitting Times to Points by a Stable Lévy Process with No Negative Jumps |
| 2. | Creator | Author's name, affiliation, country | Goran Peskir; The University of Manchester |
| 3. | Subject | Discipline(s) | |
| 3. | Subject | Keyword(s) | Stable L'evy process with no negative jumps, spectrally positive, first hitting time to a point, first passage time over a point, supremum process, a Chapman-Kolmogorov equation of Volterra type, Laplace transform, the Wiener-Hopf factorisation. |
| 3. | Subject | Subject classification | Primary 60G52, 45D05. Secondary 60J75, 45E99, 26A33. |
| 4. | Description | Abstract | Let $X=(X_t)_{t \ge 0}$ be a stable Levy process of index $\alpha \in (1,2)$ with the Levy 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: $$f_{\tau_x}(t) = \frac{x^{\alpha-1}}{\pi ( \Gamma(-\alpha) t)^{2-1/\alpha}} \sum_{n=1}^\infty \bigg[(-1)^{n-1} \sin(\pi/\alpha) \frac{\Gamma(n-1/\alpha)}{\Gamma(\alpha n-1)} \Big(\frac{x^\alpha}{c \Gamma(-\alpha)t} \Big)^{n-1} $$ $$- \sin\Big(\frac{n \pi}{\alpha}\Big) \frac{\Gamma(1+n/\alpha)}{n!} \Big(\frac{x^\alpha}{c \Gamma(-\alpha)t}\Big)^{(n+1)/\alpha-1} \bigg]$$ 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 $x<0$. |
| 5. | Publisher | Organizing agency, location | |
| 6. | Contributor | Sponsor(s) | |
| 7. | Date | (YYYY-MM-DD) | 2008-12-19 |
| 8. | Type | Status & genre | Peer-reviewed Article |
| 8. | Type | Type | |
| 9. | Format | File format | |
| 10. | Identifier | Uniform Resource Identifier | http://ecp.ejpecp.org/article/view/1431 |
| 10. | Identifier | Digital Object Identifier | 10.1214/ECP.v13-1431 |
| 11. | Source | Journal/conference title; vol., no. (year) | Electronic Communications in Probability; Vol 13 |
| 12. | Language | English=en | |
| 14. | Coverage | Geo-spatial location, chronological period, research sample (gender, age, etc.) | |
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