Passage of Lévy processes across power law boundaries at small times
Passage of Lévy processes across power law boundaries at small times
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Lévy过程在小时间跨越幂律边界
DOI：
10.1214/009117907000000097
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发表时间：
2006
影响因子：
2.3
通讯作者：
R. Maller
中科院分区：
文献类型：
--
作者：
J. Bertoin;R. Doney;R. Maller
我们希望刻画当一个Levy过程Xt在单边或双边意义上跨越tκ，κ>；0的边界时，对于小时间t；因此，我们询问LIM sup t↓0|xt|/tκ，LIM sup t↓0xt/tκ和/或LIM inf t↓0xt/tκ几乎肯定(A.S.)有限或无限。对于κ>；0的所有值，给出了这些可能性的充要条件。这是对布鲁门塔尔和格托尔在20世纪60年代的研究的补充和延伸。通常(对于κ的许多值)，当LIM SUP是有限A.S.时，它们实际上是零，但在某些情况下，LIM SUP可能具有有限的非零值A.S.一般来说，这个过程以完全不同的方式跨越单边或双边边界，但令人惊讶的是，在κ=1/2的情况下并非如此，在这种情况下，导出了一种新的具有平方根边界的重对数律的模拟。给出了一个积分检验来区分这种情况下的可能性。
We wish to characterize when a Levy process Xt crosses boundaries like tκ, κ>0, in a one- or two-sided sense, for small times t; thus, we inquire when lim sup t↓0|Xt|/tκ, lim sup t↓0Xt/tκ and/or lim inf t↓0Xt/tκ are almost surely (a.s.) finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values of κ>0. This completes and extends a line of research going back to Blumenthal and Getoor in the 1960s. Often (for many values of κ), when the lim sups are finite a.s., they are in fact zero, but the lim sups may in some circumstances take finite, nonzero, values, a.s. In general, the process crosses one- or two-sided boundaries in quite different ways, but surprisingly this is not so for the case κ=1/2, where a new kind of analogue of an iterated logarithm law with a square root boundary is derived. An integral test is given to distinguish the possibilities in that case.