α-time fractional brownian motion: PDE connections and local times
ESAIM: Probability and Statistics, Volume 16 (2012), pp. 1-24.

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.

DOI: 10.1051/ps/2011103
Classification: 60G17, 60J65, 60K99
Keywords: fractional brownian motion, strictlyα-stable Lévy process, α-time brownian motion, α-time fractional brownian motion, partial differential equation, local time, Hölder condition
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     title = {$\alpha $-time fractional brownian motion: {PDE} connections and local times},
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     pages = {1--24},
     publisher = {EDP-Sciences},
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     zbl = {1278.60074},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2011103/}
}
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Nane, Erkan; Wu, Dongsheng; Xiao, Yimin. $\alpha $-time fractional brownian motion: PDE connections and local times. ESAIM: Probability and Statistics, Volume 16 (2012), pp. 1-24. doi : 10.1051/ps/2011103. http://archive.numdam.org/articles/10.1051/ps/2011103/

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