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The first passage time problem over a moving boundary for asymptotically stable Lévy processes. (English) Zbl 1354.60049
Let \(\{X_t\}_{t\geq 0}\) be a one-dimensional Lévy process which belongs to the domain of attraction of a stable distribution with stability and skewness parameters \(\alpha\in(0,1)\) and \(\rho\in(0,1)\), respectively. Further, let \(f:[0,\infty)\to\mathbb{R}\) be the so-called moving boundary. In the paper under review, the authors study the asymptotic behavior of the first passage time of \(\{X_t\}_{t\geq0}\) over \(f(t)\), i.e., \[ \mathbb{P}(X_t\leq f(t),\;0\leq t\leq T)\quad\text{as}\quad T\to\infty. \] As the main results they show the following:
(i) If the left tail of the underlying Lévy measure has regularly varying density and \(\limsup_{t\to 0^+}\mathbb{P}(X_t\geq 0)<1\), then for any \(0<\gamma<1/\alpha\), \[ \mathbb{P}(X_t\leq 1-t^\gamma,\;0\leq t\leq T)=T^{-\rho+o(1)}\quad\text{as}\quad T\to\infty. \] (ii) If the right tail of the underlying Lévy measure has regularly varying density, then for any \(0<\gamma<1/\alpha\), \[ \mathbb{P}(X_t\leq 1+t^\gamma,\;0\leq t\leq T)=T^{-\rho+o(1)}\quad\text{as}\quad T\to\infty. \]

MSC:
60G51 Processes with independent increments; Lévy processes
60F99 Limit theorems in probability theory
60G52 Stable stochastic processes
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