Passage of Lévy processes across power law boundaries at small times

DOI10.1214/009117907000000097zbMath1140.60025arXivmath/0603274OpenAlexW3105790960MaRDI QIDQ2468425

Jean Bertoin, Ross A. Maller, Ronald Arthur Doney

Publication date: 22 January 2008

Published in: The Annals of Probability (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0603274

zbMATH Keywords

Lévy processescrossing power law boundarieslimiting and lim sup behavior.

Mathematics Subject Classification ID

Processes with independent increments; Lévy processes (60G51) Strong limit theorems (60F15) Brownian motion (60J65) Sample path properties (60G17)

Related Items (24)

Zooming in on a Lévy process at its supremum ⋮ Structure of shocks in Burgers turbulence with Lévy noise initial data ⋮ Extensions of Regularity for a Lévy Process ⋮ Intermittency in the small-time behavior of Lévy processes ⋮ Upper functions for sample paths of Lévy(-type) processes ⋮ Short-time behavior of solutions to Lévy-driven stochastic differential equations ⋮ How smooth can the convex hull of a Lévy path be? ⋮ Small time Chung-type LIL for Lévy processes ⋮ Small and large time stability of the time taken for a Lévy process to cross curved boundaries ⋮ Hölder continuity of the convex minorant of a Lévy process ⋮ Zooming-in on a Lévy process: failure to observe threshold exceedance over a dense grid ⋮ Finite variation of fractional Lévy processes ⋮ Weighted empirical processes in the nonparametric inference for Lévy processes ⋮ Small-time compactness and convergence behavior of deterministically and self-normalised Lévy processes ⋮ An almost sure functional limit theorem at zero for a class of Lévy processes normed by the square root function, and applications ⋮ Small time one-sided LIL behavior for Lévy processes at zero ⋮ The small-time Chung-Wichura law for Lévy processes with non-vanishing Brownian component ⋮ Ruin and deficit under claim arrivals with the order statistics property ⋮ Conditions for a Lévy process to stay positive near 0, in probability ⋮ Small time two-sided LIL behavior for Lévy processes at zero ⋮ LIL type behavior of multivariate Lévy processes at zero ⋮ Laws of the iterated logarithm for self-normalised Lévy processes at zero ⋮ A lifetime of excursions through random walks and Lévy processes ⋮ On multivariate quasi-infinitely divisible distributions

Cites Work

• The lower limit of a normalized random walk
• The growth of random walks and Levy processes
• Regularity of the half-line for Lévy processes
• Stability and attraction to normality for Lévy processes at zero and at infinity
• Stability of the overshoot for Lévy processes
• Passage times of random walks and Lévy processes across power law boundaries
• Finiteness of integrals of functions of Lévy processes
• Local Behavior of Processes with Independent Increments
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