Heavy tails of a Lévy process and its maximum over a random time interval. (English) Zbl 1236.60046
Summary: Let \(\{X _{t },\;t \geqslant 0\}\) be a Lévy process with Lévy measure \(\nu \) on \(( - \infty ,\infty )\), and let \(\tau \) be a nonnegative random variable independent of \(\{X _{t }, t \geqslant 0\}\). We are interested in the tail probabilities of \(X _{\tau }\) and \(X _{(\tau )} = \sup_{0\leqslant t \leqslant \tau } X _{t }\). For various cases, under the assumption that either the Lévy measure \(\nu \) or the random variable \(\tau \) have a heavy right tail, we prove that both Pr\((X _{\tau } > x)\) and Pr\((X _{(\tau )} > x)\) are asymptotic to \(\operatorname{E}\tau \nu ((x, \infty ))\) + Pr\((\tau > x/(0 \vee \operatorname{E}X _{1}))\) as \(x \rightarrow \infty \), where Pr\((\tau > x/0) = 0\) by convention.
MSC:
| 60G51 | Processes with independent increments; Lévy processes |
| 60E07 | Infinitely divisible distributions; stable distributions |
| 60F10 | Large deviations |
| 60G70 | Extreme value theory; extremal stochastic processes |
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