Unilateral Small Deviations for the Integral of Fractional Brownian Motion

Abstract

We consider the paths of a Gaussian random process $x(t)$, $x(0)=0$ not exceeding a fixed positive level over a large time interval $(0,T)$, $T\gg 1$. The probability $p(T)$ of such event is frequently a regularly varying function at $\infty$ with exponent $\theta$. In applications this parameter can provide information on fractal properties of processes that are subordinate to $x(\cdot)$. For this reason the estimation of $\theta$ is an important theoretical problem. Here, we consider the process $x(t)$ whose derivative is fractional Brownian motion with self-similarity parameter $0<H<1$. For this case we produce new computational evidence in favor of the relations $\log p(T)=-\theta \log T(1+o(1))$ and $\theta =H(1-H)$. The estimates of $\theta$ are to within 0.01 in the range $0.1\le H\le 0.9$. An analytical result for the problem in hand is known for the markovian case alone, i.e., for $H=1/2$. We point out other statistics of $x(t)$ whose small values have probabilities of the same order as $p(T)$ in the $\log$ scale.