Upper functions for sample paths of L\'evy(-type) processes

DOI10.3150/21-BEJ1441arXiv2102.06541WikidataQ113701724 ScholiaQ113701724MaRDI QIDQ6360463

Publication date: 12 February 2021

Abstract: We study the small-time asymptotics of sample paths of L'evy processes and L'evy-type processes. Namely, we investigate under which conditions the limit

limsup_{t o 0} frac{1}{f(t)} |X_t-X_0|

is finite resp. infinite with probability

1

. We establish integral criteria in terms of the infinitesimal characteristics and the symbol of the process. Our results apply to a wide class of processes, including solutions to L'evy-driven SDEs and stable-like processes. For the particular case of L'evy processes, we recover and extend earlier results from the literature. Moreover, we present a new maximal inequality for L'evy-type processes, which is of independent interest.

Mathematics Subject Classification ID

Processes with independent increments; Lévy processes (60G51) Martingales with continuous parameter (60G44) Stable stochastic processes (60G52) Feller processes (60G53)

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