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

  • Erkan Nane (a1), Dongsheng Wu (a2) and Yimin Xiao (a3)
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.

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.

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ESAIM: Probability and Statistics
  • ISSN: 1292-8100
  • EISSN: 1262-3318
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