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

TL;DR

This paper establishes the existence, uniqueness, and growth bounds of solutions to rough differential equations driven by tempered fractional Brownian motion with Hurst index between 1/4 and 1/3, using rough path theory and transformations.

Contribution

It introduces a novel approach to handle rough differential equations driven by TFBM with low Hurst index, including a canonical lift and a transformation technique for solution analysis.

Findings

Constructed a canonical lift of TFBM to a geometric rough path.

Proved existence and uniqueness of solutions to the RDE driven by TFBM.

Derived an upper bound for the solution norm using Gronwall's lemma.

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 $\lambda>0$. 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.