Convergence rate in the splitting-up method for rough differential equations

TL;DR

The paper addresses convergence and rate analysis for a splitting-up numerical scheme applied to rough differential equations driven by a Hölder path $X$ with $\alpha\in(1/3,1/2]$, constructing solutions via a two-step operator-splitting that incorporates the second-order area information $\mathbb{X}_{s,t}$ and a correction operator $Z$. It embeds the problem in the rough-path framework with the augmented driver $\mathbf{X}=(X,\mathbb{X})$ and uses a sewing-like argument to show that the splitting approximation converges to a Davie-sense solution, providing explicit convergence rates in $C^\alpha$ and $C^\beta$ spaces. The results rely on analytic and algebraic conditions on $Z$ that control the size, Hölder continuity, and coboundary structure of the second-order term, and they establish a precise rate for the dyadic refinement $h=2^{-n}$, together with a refined bound for rational time-step ratios. Overall, the work links explicit splitting schemes to robust notions of RDE solutions and yields quantitative convergence, extending classical Milstein-type ideas to rough path settings with a clear path to generalizations and implicit schemes in future work.

Abstract

In this note we construct solutions to rough differential equations ${\rm d} Y = f(Y) \,{\rm d} X$ with a driver $X \in C^α([0,T];\mathbb{R}^d)$, $\frac13 < α\le \frac12$, using a splitting-up scheme. We show convergence of our scheme to solutions in the sense of Davie by a new argument and give a rate of convergence.

Figures (1)

• Figure 1: Schematic plot of the splitting scheme, with vertical traversal signifying evolution by the first equation, and horizontal traversal signifying evolution under the second equation.

Theorems & Definitions (12)

• Definition 1.1
• Proposition 2.1
• proof
• Corollary 2.2
• proof
• Proposition 3.1
• Lemma 3.2
• Remark 3.1
• proof
• Lemma 3.3