Asymptotic expansion of the weighted power variation with second order differences of a stochastic differential equation driven by fBm

Abstract: We study a process satisfying a one-dimensional stochastic differential equation driven by fractional Brownian motion with Hurst index $H>1/2$, and consider the weighted power variation based on the second order differences of the process. We derive the asymptotic expansion formula of its distribution based on the theory of expansion of Skorohod integrals by Nualart and Yoshida. The formula includes the rate of convergence as a corollary. To facilitate the application of the general expansion theory, we employ the theory of exponents from arXiv:2407.02254 to obtain estimates of functionals.