Passage of Lévy Processes across Power Law Boundaries at Small Times

We wish to characterize when a Lévy process Xt crosses boundaries like t�, �>0, in a oneor two-sided sense, for small times t; thus, we inquire when lim sup t↓0|Xt|/t�, lim sup t↓0Xt/t� and/or lim inf t↓0Xt/t� are almost surely (a.s.) finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values of �>0. This completes and extends a line of research going back to Blumenthal and Getoor in the 1960s. Often (for many values of �), 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 oneor two-sided boundaries in quite different ways, but surprisingly this is not so for the case �=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. DOI: https://doi.org/10.1214/009117907000000097 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-78181 Published Version Originally published at: 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. DOI: https://doi.org/10.1214/009117907000000097 The Annals of Probability 2008, Vol. 36, No. 1, 160–197 DOI: 10.1214/009117907000000097 © Institute of Mathematical Statistics, 2008 PASSAGE OF LÉVY PROCESSES ACROSS POWER LAW BOUNDARIES AT SMALL TIMES1 BY J. BERTOIN, R. A. DONEY AND R. A. MALLER Université Pierre et Marie Curie, University of Manchester and Australian National University We wish to characterize when a Lévy process Xt crosses boundaries like tκ , κ > 0, in a oneor two-sided sense, for small times t ; thus, we inquire when lim supt↓0 |Xt |/tκ , lim supt↓0 Xt/t and/or lim inft↓0 Xt/t 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 oneor 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.