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 $m\to\infty$. Although we cannot use martingale central limit theorem, the fourth moment theorem helps us and we already have useful limit theorems of weighted sum processes of Wiener chaos and they can be applied to the study of the asymptotic error distribution. In fact, for some typical approximate solutions, it is 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 $B_t$. One of our main results is as follows. The difference $R^m_t=\hat{Y}^m_t-Y_t-J_tI^m_t$ is really small compared to the main term $J_tI^m_t$. That is, we show that $(2^m)^{2H-1/2+\varepsilon}\sup_{0\le t\le 1}|R^m_t|\to 0$ almost surely and in $L^p$ (for all $p\>1$) for certain explicit positive number $\varepsilon>0$. To this end, 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.