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

TL;DR

This paper demonstrates that the tail probability of certain rough line integrals driven by fractional Brownian motion with Hurst parameter between 1/4 and 1/2 cannot decay faster than a Weibull tail, showing solutions to related differential equations lack Gaussian tails.

Contribution

It provides the first explicit tail behavior lower bound for rough line integrals along fractional Brownian paths, confirming the sharpness of previous upper tail estimates.

Findings

Tail probability cannot decay faster than a Weibull tail with exponent > 2H+1.

Solutions to certain fractional Brownian-driven differential equations lack Gaussian tails.

Validates the sharpness of prior upper tail bounds by Cass-Litterer-Lyons.

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.