Fine regularity of Lévy processes and linear (multi)fractional stable motion. (English) Zbl 1307.60055

Balança, Paul

Electron. J. Probab. 19, Paper No. 101, 37 p. (2014).

Summary: In this work, we investigate the fine regularity of Lévy processes using the 2-microlocal formalism. This framework allows us to refine the multifractal spectrum determined by S. Jaffard [Probab. Theory Relat. Fields 114, No. 2, 207–227 (1999; Zbl 0947.60039)] and, in addition, study the oscillating singularities of Lévy processes. The fractal structure of the latter is proved to be more complex than the classic multifractal spectrum and is determined in the case of alpha-stable processes. As a consequence of these fine results and the properties of the 2-microlocal frontier, we are also able to completely characterise the multifractal nature of the linear fractional stable motion (extension of fractional Brownian motion to \(\alpha \)-stable measures) in the case of continuous and unbounded sample paths as well. The regularity of its multifractional extension is also presented, indirectly providing an example of a stochastic process with a non-homogeneous and random multifractal spectrum.

MSC:

60G51	Processes with independent increments; Lévy processes
60G07	General theory of stochastic processes
60G17	Sample path properties
60G22	Fractional processes, including fractional Brownian motion
60G44	Martingales with continuous parameter

Keywords:

Lévy processes; linear fractional stable motion; 2-microlocal analysis; Hölder regularity; multifractal spectrum; oscillating singularities

Citations:

Zbl 0947.60039