Optimal Tail Estimates for Differential Equations Driven by Fractional Brownian Motion
Abstract
The goal of the present paper is to investigate the exact decay rate of the tail probability ℙ(|X1−x0|>R) for large R, where Xt is the solution to a multidimensional stochastic differential equation driven by a fractional Brownian motion with initial condition x0. In the first place, under the assumption of uniform ellipticity, we establish a general (2H+1)-Weibull lower tail estimate in Young's regime of H∈(1/2,1) and a general Gaussian lower tail estimate in the rough regime of H∈(1/4,1/2). These two estimates are seen to be sharp in their respective regimes. In the second place, we prove a striking fact by constructing explicit examples that a uniformly elliptic system with Cb∞-coefficients could \textit{fail} to have Gaussian lower tail in Young's regime. As a consequence, in our modest opinion, the multidimensional Gaussian lower estimate of \cite{BKT16} might not hold in its current form of generality without further assumptions on the vector fields. In the third place, we provide a simple nondegeneracy condition on the vector fields, under which a Gaussian lower tail can be established in Young's regime. In addition, given the existence of the aforementioned counterexamples, we prove another striking fact that any 2× 2 periodic, uniformly elliptic system always has Gaussian lower tail in Young's regime. Lastly, in the rough regime we establish a general (2H+1)-Weibull lower tail for a rich class of systems that satisfy a noncommutativity condition on the vector fields. This result provides a partially affirmative answer to a conjecture raised in \cite{BG24} on the genericness of the well-known Cass-Litterer-Lyons estimate for noncommutative systems.
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BibTeX
@misc{boedihardjo2026optimal,
title = {{Optimal Tail Estimates for Differential Equations Driven by Fractional Brownian Motion}},
author = {Horatio Boedihardjo and Xi Geng and Sheng Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06152},
eprint = {2610.06152},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06152},
note = {CC0},
}
RIS
TY - UNPB
TI - Optimal Tail Estimates for Differential Equations Driven by Fractional Brownian Motion
AU - Horatio Boedihardjo
AU - Xi Geng
AU - Sheng Wang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06152
UR - https://arxiv.org/abs/2610.06152
AB - The goal of the present paper is to investigate the exact decay rate of the tail probability ℙ(|X_1−x_0|>R) for large R, where X_t is the solution to a multidimensional stochastic differential equation driven by a fractional Brownian motion with initial condition x_0. In the first place, under the assumption of uniform ellipticity, we establish a general (2H+1)-Weibull lower tail estimate in Young's regime of H∈(1/2,1) and a general Gaussian lower tail estimate in the rough regime of H∈(1/4,1/2). These two estimates are seen to be sharp in their respective regimes. In the second place, we prove a striking fact by constructing explicit examples that a uniformly elliptic system with C_b^∞-coefficients could \textit{fail} to have Gaussian lower tail in Young's regime. As a consequence, in our modest opinion, the multidimensional Gaussian lower estimate of \cite{BKT16} might not hold in its current form of generality without further assumptions on the vector fields. In the third place, we provide a simple nondegeneracy condition on the vector fields, under which a Gaussian lower tail can be established in Young's regime. In addition, given the existence of the aforementioned counterexamples, we prove another striking fact that any 2× 2 periodic, uniformly elliptic system always has Gaussian lower tail in Young's regime. Lastly, in the rough regime we establish a general (2H+1)-Weibull lower tail for a rich class of systems that satisfy a noncommutativity condition on the vector fields. This result provides a partially affirmative answer to a conjecture raised in \cite{BG24} on the genericness of the well-known Cass-Litterer-Lyons estimate for noncommutative systems.
ER -