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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.

60J99 Markov processes
60G17 Sample path properties
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