On the lack of Gaussian tail for rough line integrals along fractional Brownian paths

• 成果类型: Article
• 署名作者: Boedihardjo, H.; Geng, X.
• 署名单位: University of Warwick; University of Melbourne
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01242-4
• 发表日期: 2024
• 页码: 1287-1313
• 关键词: differential-equations driven

摘要

We show that the tail probability of the rough line integral f(0)(1) phi(X-t)dY(t), where (X, Y) is a 2D fractional Brownian motion with Hurst parameter H is an element of (1/4, 1/2) and phi is a C-b(infinity)-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(infinity). This also demonstrates that the well-known upper tail estimate proved by Cass-Litterer-Lyons in 2013 is essentially sharp.