On the first passage of one-sided Lévy motions

✍ Scribed by Iddo Eliazar; Joseph Klafter

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 341 KB
• Volume: 336
• Category: Article
• ISSN: 0378-4371
• DOI: 10.1016/j.physa.2003.12.032

✦ Synopsis

We study the ÿrst passage problem for one-sided LÃ evy motions: how does such a motion, started at the origin, cross a barrier positioned at the point x (x ¿ 0)? Since one-sided LÃ evy motions are pure-jump processes, they always 'leap' over barriers (rather than crossing them continuously). We hence explore the following issues: (i) ÿrst passage times (FPTs)-how long would it take the motion to cross the barrier; (ii) ÿrst passage leapovers (FPLs)-how far would the motion leap over the barrier ; and, (iii) what is dependence between the FPTs and the FPLs.

Formulae for the joint Laplace transforms of the FPTs and FPLs are derived, and the scaling limits of the FPTs and FPLs are computed. The scaling limits turn out to display dramatically di erent behavior when: (i) the underlying LÃ evy motion has ÿnite cumulants; and when, (ii) the underlying LÃ evy motion is heavy-tailed. Special attention is devoted to the investigation of the ÿrst passage problem for self-similar one-sided LÃ evy motions.