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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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References:
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