An interpolation of discrete rough differential equations and its applications to analysis of error distributions

TL;DR

This paper develops an interpolation framework to analyze the asymptotic error distribution of approximate solutions to rough differential equations driven by fractional Brownian motion, providing new convergence rate results.

Contribution

It introduces an interpolation process that simplifies the analysis of error distribution in rough differential equations, strengthening previous results.

Findings

The difference between the approximate and true solution is asymptotically negligible compared to the main error term.

The convergence rate of the approximation error is explicitly estimated in both almost sure and $L^p$ senses.

The interpolation approach offers a new tool for analyzing error distributions in rough path theory.

Abstract

We consider the solution $Y_t$ $(0\le t\le 1)$ and several approximate solutions $\hat{Y}^m_t$ of a rough differential equation driven by a fractional Brownian motion $B_t$ with the Hurst parameter $1/3<H\leq 1/2$ associated with a dyadic partition of $[0,1]$. We are interested in analysis of asymptotic error distribution of $\hat{Y}^m_t-Y_t$ as $m\to\infty$. In the preceding results, it was proved that the weak limit of $\{(2^m)^{2H-1/2}(\hat{Y}^m_t-Y_t)\}_{0\le t\le 1}$ coincides with the weak limit of $\{(2^m)^{2H-1/2}J_tI^m_t\}_{0\le t\le 1}$, where $J_t$ is the Jacobian process of $Y_t$ and $I^m_t$ is a certain weighted sum process of Wiener chaos of order $2$ defined by $B_t$. However, it is non-trivial to reduce a problem about $\hat{Y}^m_t-Y_t$ to one about $J_t$ and $I^m_t$. In this paper, we introduce an interpolation process between $Y_t$ and $\hat{Y}^m_t$, and give several estimates of the interpolation process itself and its associated processes. The analysis provides a framework to deal with the reduction problem and provides a stronger result that the difference $R^m_t=\hat{Y}^m_t-Y_t-J_tI^m_t$ is really small compared to the main term $J_tI^m_t$. More precisely, we show that $(2^m)^{2H-1/2+\varepsilon}\sup_{0\leq t\leq 1}|R^m_t|\to 0$ almost surely and in $L^p$ (for all $p>1$) for certain explicit positive number $\varepsilon>0$. As a consequence, we obtain an estimate of the convergence rate of $\sup_{0\leq t\leq 1}|\hat{Y}^m_t-Y_t|\to 0$ in $L^p$ also.