Consider a diffusion process \(X = (X_t)_{t \in [0,T]}, ~ T > 0\) , solution of a time-homogeneous stochastic differential equation observed at discrete times and high frequency. We assume that the drift and diffusion coefficients of the process X are unknown. In this paper, we study the minimax convergence rates of projection estimators of the square of the diffusion coefficient. Two observation schemes are considered depending on the estimation interval. The square of the diffusion coefficient is estimated on the real line from repeated observations of the process X, where the number of diffusion paths tends to infinity. For the case of a compact estimation interval, we study projection estimators of the square of the diffusion coefficient constructed from a single diffusion path on one side and from repeated observations on the other side, where the number of trajectories tends to infinity. In each of these cases, we establish minimax convergence rates of the worst risk of estimation over a space of Hölder functions.
