Asymptotics for first-passage times of L\'evy processes and random walks

Denis Denisov; Vsevolod Shneer

Asymptotics for first-passage times of L\'evy processes and random walks

Probability 2007-12-06 v1

Authors: Denis Denisov , Vsevolod Shneer

Abstract

We study the exact asymptotics for the distribution of the first time $\tau_x$ a L\'evy process $X_t$ crosses a negative level $-x$ . We prove that $\mathbf P(\tau_x>t)\sim V(x)\mathbf P(X_t\ge 0)/t$ as $t\to\infty$ for a certain function $V(x)$ . Using known results for the large deviations of random walks we obtain asymptotics for $\mathbf P(\tau_x>t)$ explicitly in both light and heavy tailed cases. We also apply our results to find asymptotics for the distribution of the busy period in an M/G/1 queue.

Keywords

Lévy processes first-passage times asymptotics random walks large deviations M/G/1 queue

Cite

@article{arxiv.0712.0728,
  title  = {Asymptotics for first-passage times of L\'evy processes and random walks},
  author = {Denis Denisov and Vsevolod Shneer},
  journal= {arXiv preprint arXiv:0712.0728},
  year   = {2007}
}

Asymptotics for first-passage times of L\'evy processes and random walks

Abstract

Keywords

Cite

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