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α-time fractional Brownian motion: PDE connections and local times

Published online by Cambridge University Press:  09 March 2012

Erkan Nane
Affiliation:
Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL 36849, USA. www.duc.auburn.edu/˜ezn0001/. nane@stt.msu.edu
Dongsheng Wu
Affiliation:
Department of Mathematical Sciences, 201J Shelby Center, University of Alabama in Huntsville, Huntsville, AL 35899, USA; http://webpages.uah.edu/˜dw0001. dongsheng.wu@uah.edu
Yimin Xiao
Affiliation:
Department of Statistics and Probability, A-413 Wells Hall, Michigan State University, East Lansing, MI 48824, USA; http://www.stt.msu.edu/˜xiaoyimi. xiao@stt.msu.edu
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Abstract

For 0 < α ≤ 2 and 0 < H < 1, an α-time fractional Brownian motion is an iterated process Z =  {Z(t) = W(Y(t)), t ≥ 0}  obtained by taking a fractional Brownian motion  {W(t), t ∈ ℝ} with Hurst index 0 < H < 1 and replacing the time parameter with a strictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ R}. It is shown that such processes have natural connections to partial differential equations and, when Y is a stable subordinator, can arise as scaling limit of randomly indexed random walks. The existence, joint continuity and sharp Hölder conditions in the set variable of the local times of a d-dimensional α-time fractional Brownian motion X = {X(t), t ∈ ℝ+} defined by X(t) = (X1(t), ..., Xd(t)), where t ≥ 0 and X1, ..., Xd are independent copies of Z, are investigated. Our methods rely on the strong local nondeterminism of fractional Brownian motion.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2012

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