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Passage of Lévy Processes across Power Law Boundaries at Small Times

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

Description

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]

dc.contributor.authorBertoin, Jean
dc.contributor.authorDoney, Ronald A.
dc.contributor.authorMaller, Ross A.
dc.date.accessioned2016-03-06T23:15:18Z
dc.date.available2016-03-06T23:15:18Z
dc.identifier.issn0091-1798
dc.identifier.urihttp://hdl.handle.net/1885/100170
dc.description.abstractWe 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 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.
dc.description.sponsorshipSupported in part by ARC Grants DP0210572 and DP0664603.
dc.publisherInstitute of Mathematical Statistics
dc.rights© Institute of Mathematical Statistics, 2008. http://www.sherpa.ac.uk/romeo/issn/0091-1798..."author can archive publisher's version/PDF. On author's personal website or open access repository" from SHERPA/RoMEO site (as at 7/03/16).
dc.sourceThe Annals of Probability
dc.subjectLévy processes
dc.subjectcrossing power law boundaries
dc.subjectlimiting and lim sup behavior
dc.titlePassage of Lévy Processes across Power Law Boundaries at Small Times
dc.typeJournal article
local.description.notesImported from ARIES
local.identifier.citationvolume36
dc.date.issued2008
local.identifier.absfor010404
local.identifier.ariespublicationu4085724xPUB60
local.publisher.urlhttp://imstat.org/en/index.html
local.type.statusPublished Version
local.contributor.affiliationBertoin, Jean, Universite Paris VI, France
local.contributor.affiliationDoney, R A, University of Manchester, United Kingdom
local.contributor.affiliationMaller, Ross, College of Physical and Mathematical Sciences, CPMS Mathematical Sciences Institute, Centre for Mathematics and Its Applications, The Australian National University
dc.relationhttp://purl.org/au-research/grants/arc/DP0210572
dc.relationhttp://purl.org/au-research/grants/arc/DP0664603
local.bibliographicCitation.issue1
local.bibliographicCitation.startpage160
local.bibliographicCitation.lastpage197
local.identifier.doi10.1214/009117907000000097
dc.date.updated2016-06-14T08:58:52Z
local.identifier.scopusID2-s2.0-39449092987
local.identifier.thomsonID000251754100006
dcterms.accessRightsOpen Access
CollectionsANU Research Publications

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