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α-time fractional Brownian motion: PDEconnections and local times∗

Published online by Cambridge University Press:  09 March 2012

Erkan Nane ,
Dongsheng Wu  and
Yimin Xiao
Erkan Nane
Affiliation:
Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL 36849, USA. www.duc.auburn.edu/˜ezn0001/. [email protected]
Dongsheng Wu
Affiliation:
Department of Mathematical Sciences, 201J Shelby Center, University of Alabama in Huntsville, Huntsville, AL 35899, USA; http://webpages.uah.edu/˜dw0001. [email protected]
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. [email protected]
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Abstract

For 0 < α ≤ 2 and 0 < H < 1, anα-time fractional Brownian motion is an iterated processZ =  {Z(t) = W(Y(t)), t ≥ 0} obtained by taking a fractional Brownian motion {W(t), t ∈ ℝ} with Hurst index0 < H < 1 and replacing the time parameter with astrictly α-stable Lévy process {Y(t), t ≥ 0} in ℝ independent of {W(t), t ∈ R}. It is shown that suchprocesses have natural connections to partial differential equations and, whenY is a stable subordinator, can arise as scaling limit of randomlyindexed random walks. The existence, joint continuity and sharp Hölder conditions in theset variable of the local times of a d-dimensionalα-time fractional Brownian motionX = {X(t), t ∈ ℝ+} defined by X(t) = (X1(t), ...,Xd(t)), where t ≥ 0 andX1, ..., Xdare independent copies of Z, are investigated. Our methods rely on thestrong local nondeterminism of fractional Brownian motion.

Keywords

Fractional Brownian motionstrictlyα-stable Lévy processα-time Brownian motionα-time fractional Brownian motionpartial differential equationlocal timeHölder condition.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 1 - 24
Copyright
© EDP Sciences, SMAI, 2012

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