Passage of Lévy processes across power law boundaries at small times

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 th Show more