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PATH PROPERTIES OF THE LINEAR MULTIFRACTIONAL STABLE MOTION

STILIAN STOEV

Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, USA

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and

MURAD S. TAQQU

Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, USA

Corresponding author.

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https://doi.org/10.1142/S0218348X05002775Cited by:27 (Source: Crossref)

Abstract

The linear multifractional stable motion (LMSM) processes Y = {Y(t)}_t∈ℝ is an α-stable (0 < α < 2) stochastic process, which exhibits local self-similarity, has heavy tails and can have skewed distributions. The process Y is obtained from the well-known class of linear fractional stable motion (LFSM) processes by replacing their self-similarity parameter H by a function of time H(t).

We show that the paths of Y(t) are bounded on bounded intervals only if 1/α ≤ H(t) < 1, t ∈ ℝ. In particular, if 0 < α ≤ 1, then Y has everywhere discontinuous paths, with probability one. On the other hand, Y has a version with continuous paths if H(t) is sufficiently regular and 1/α < H(t), t ∈ ℝ. We study the Hölder regularity of the sample paths when these are continuous and establish almost sure bounds on the pointwise and uniform pointwise Hölder exponents of the (random) function Y(t,ω), t ∈ ℝ, in terms of the function H(t) and its corresponding Hölder exponents.

The Gaussian multifractional Brownian motion (MBM) processes are LMSM processes when α = 2. We obtain some new results on the Hölder regularity of their paths.

Keywords:

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