Nonparametric estimation of scalar diffusions based on low frequency data is ill-posed

We study the problem of estimating the coefficients of a diffusion (Xt, t ≥ 0); the estimation is based on discrete data Xn,n = 0,1,...,N. The sampling frequency −1 is constant, and asymptotics are taken as the number N of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient (the volatility) and the drift in a nonparametric setting is ill-posed: the minimax rates of convergence for Sobolev constraints and squared-error loss coincide with that of a, respectively, first- and second-order linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions. Our approach is based on the spectral analysis of the associated Markov semigroup. A rate-optimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue–eigenfunction pair of the transition operator of the discrete time Markov chain (Xn,n = 0,1,...,N) in a suitable Sobolev norm, together with an estimation of its invarian