On the Lack of Gaussian Tail for Rough Line Integrals along Fractional Brownian Paths

Abstract

We show that the tail probability of the rough line integral $\int_{0}^{1}\phi(X_{t})dY_{t}$, where $(X,Y)$ is a 2D fractional Brownian motion with Hurst parameter $H\in(1/4,1/2)$ and $\phi$ is a $C_{b}^{\infty}$-function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a $\gamma$-Weibull tail with any exponent $\gamma>2H+1$. In particular, this produces a simple class of examples of differential equations driven by fBM, whose solutions fail to have Gaussian tail even though the underlying vector fields are assumed to be of class $C_{b}^{\infty}$. This also demonstrates that the well-known upper tail estimate proved by Cass-Litterer-Lyons in 2013 is essentially sharp.