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

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.