Large deviations for local time fractional Brownian motion and applications

Abstract

Let $W^H=\{W^H(t), t \in \rr\}$ be a fractional Brownian motion of Hurst index $H\in (0,1)$ with values in $\rr$, and let $L=\{L_t,t\ge 0\}$ be the local time process at zero of a strictly stable Lévy process $X=\{X_t,t\ge 0\}$ of index $1<\alpha \le 2$ independent of $W^H$. The $\a$-stable local time fractional Brownian motion $Z^H=\{Z^H(t),t\ge 0\}$ is defined by $Z^H(t)=W^H(L_t)$. The process $Z^H$ is self-similar with self-similarity index $H(1-\frac{1}{\alpha })$ and is related to the scaling limit of a continuous time random walk with heavy-tailed waiting times between jumps (\cite{coupleCTRW,limitCTRW}). However, $Z^H$ does not have stationary increments and is non-Gaussian. In this paper we establish large deviation results for the process $Z^H$. As applications we derive upper bounds for the uniform modulus of continuity and the laws of the iterated logarithm for $Z^H$.

20 pages

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Cited by in corpus (6)

• Correlated continuous time random walks
• Space-time duality for fractional diffusion
• Laws of the iterated logarithm for a class of iterated processes
• Exact Asymptotic Formulas for the Heat Kernels of Space and Time-Fractional Equations
• Harnack Inequalities for SDEs Driven by Time-Changed Fractional Brownian Motions
• Asymptotic properties of Brownian motion delayed by inverse subordinators