[论文解读] Rough differential equations driven by TFBM with Hurst index $H\in (\frac{1}{4}, \frac{1}{3})$

Lijuan Zhang, Jianhua Huang|arXiv (Cornell University)|Mar 8, 2026

Stochastic processes and financial applications被引用 0

一句话总结

该论文证明了由 tempered fractional Brownian motion 驱动的粗糙微分方程在 H∈(1/4,1/3) 时的存在性与唯一性，采用三步几何粗糙路径提升与 Doss-Sussmann 变换结合贪心停止时间，以及对解的先验界限。

ABSTRACT

We consider the rough differential equations driven by tempered fractional Brownian motion with Hurst index $H\in (\frac{1}{4}, \frac{1}{3})$ and tempered parameter $λ>0$. First, by means of piecewise linear approximation, we canonically lift the tempered fractional Brownian motion to a three-step geometric rough path in an almost sure sense. Subsequently, employing the Doss-Sussmann technique in conjunction with a greedy sequence of stopping times, we construct a suitable transformation that establishes a bijection between the solution of the rough differential equation and that of an associated ordinary differential equation. This yields the existence and uniqueness of a solution to the original equation. Based on this result and appealing to Gronwall's lemma, we further derive an upper bound for the solution norm, thereby providing a quantitative control on its growth.

研究动机与目标

• Motivate the study of rough differential equations driven by tempered fractional Brownian motion (TFBM) with H in (1/4,1/3).
• Construct a canonical three-step geometric rough path lift for TFBM via piecewise linear approximation.
• Establish existence and uniqueness of solutions to the rough differential equation using Doss-Sussmann transformation and greedy stopping times.
• Obtain quantitative growth control for the solution through Gronwall-type estimates.

提出的方法

• Lift TFBM to a three-step geometric rough path using piecewise linear approximation and convergence arguments.
• Use covariance structure of TFBM and properties of the Bessel function to handle divergences in the H in (1/4,1/3) regime.
• Apply the Doss-Sussmann technique to transform the rough differential equation into an associated ordinary differential equation.
• Prove local existence/uniqueness for the transformed ODE on a small interval and extend globally via a greedy sequence of stopping times.
• Concatenate local solutions on subintervals to obtain a global solution and derive an upper bound for the solution norm via Gronwall’s lemma.

实验结果

研究问题

• RQ1 Can a rough differential equation driven by tempered fractional Brownian motion with H in (1/4,1/3) be given a well-posed solution concept?
• RQ2 Does a canonical rough path lift exist for tempered fractional Brownian motion in this Hurst range?
• RQ3 Can the Doss-Sussmann transformation be employed to reduce the rough equation to an ODE and ensure existence/uniqueness?
• RQ4 How can greedy stopping times be used to extend local solutions to global ones on arbitrary intervals?
• RQ5 What quantitative bound can be established on the solution growth?

主要发现

• A canonical lift of tempered fractional Brownian motion to a third-level geometric rough path is constructed almost surely via piecewise linear approximation.
• A bijection between solutions of the rough differential equation and an associated ordinary differential equation is established through a Doss-Sussmann transformation.
• Local existence and uniqueness of the rough equation is proved under global Lipschitz conditions on f and sufficient regularity on g.
• A greedy sequence of stopping times extends the local solution to a global one on arbitrary intervals.
• An upper bound for the solution norm is derived using Gronwall’s lemma, yielding quantitative growth control.