# Passage of Lévy processes across power law boundaries at small times

Bertoin, Jean; Doney, R; Maller, R (2008). Passage of Lévy processes across power law boundaries at small times. The Annals of Probability, 36(1):160-197.

## Abstract

We wish to characterize when a Lévy 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.

We wish to characterize when a Lévy 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.

## Citations

15 citations in Web of Science®
16 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2008 29 May 2013 15:18 05 Apr 2016 16:47 Institute of Mathematical Statistics 0091-1798 Publisher DOI. An embargo period may apply. https://doi.org/10.1214/009117907000000097 http://www.ams.org/mathscinet-getitem?mr=2370602http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1140.60025
Permanent URL: https://doi.org/10.5167/uzh-78181

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