Large deviations for local time fractional Brownian motion and applications

TL;DR

This paper establishes large deviation principles for the non-Gaussian, self-similar local time fractional Brownian motion, providing insights into its regularity and asymptotic behaviors.

Contribution

It introduces large deviation results for the process Z^H, a novel non-Gaussian, self-similar process constructed from fractional Brownian motion and local time of a stable Lévy process.

Findings

Large deviation principles for Z^H are proved.

Upper bounds for the modulus of continuity are derived.

Laws of the iterated logarithm for Z^H are established.

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 L\'evy process $X=\{X_t, t \ge 0\}$ of index $1<\alpha\leq 2$ 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 $Z^H$.