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On the first exit time of a completely asymmetric stable process from a finite interval. (English) Zbl 0863.60068
The paper is concerned with the distribution of the first exit time $$T_{a,b}$$ from a finite interval $$[-a,b]$$ for a completely asymmetric stable process $$X_t$$ with exponent $${\mathbf E}(\exp\{\lambda X_t\})=\exp\{\lambda^\alpha\}$$, $$t,\lambda \geq 0$$, $$\alpha\in (1,2]$$. There are numerous works related to this functional, but the distinctive feature of this one consists in using Mittag-Leffler function and its derivative to obtain the explicit formula for the characteristic function of $$T_{a,b}$$. It looks as follows ${\mathbf E}\exp\{-qT_{a,b}\}=E_\alpha(qa^\alpha)-\left(a\over a+b\right)^{\alpha - 1}{E_\alpha'(qa^\alpha)\over E_\alpha'(q(a+b)^\alpha)}\{E_\alpha(q(a+b)^\alpha)-1\}$ where $$E_\alpha(x)=\sum_{n=0}^\infty x^n/\Gamma(1+\alpha n)$$, $$x\in \mathbb{R}$$, denotes the Mittag-Leffler function of the parameter $$\alpha$$. As an application of this formula, the asymptotic tail behavior of the distribution in question and the law of the iterated logarithm have been considered as well.

##### MSC:
 60J99 Markov processes 60G17 Sample path properties
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