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Passage of Lévy Processes across Power Law Boundaries at Small Times

Bertoin, Jean; Doney, Ronald A.; Maller, Ross A.


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.supt↓0 |Xt|/tκ, lim supt↓0, Xt/tκ and/or lim inft↓0 Xt/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...[Show more]

CollectionsANU Research Publications
Date published: 2008
Type: Journal article
Source: The Annals of Probability
DOI: 10.1214/009117907000000097


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