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 $1/3 1$) 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.