Nonparametric estimation of the diffusion coefficient from i.i.d. S.D.E. paths

Consider a diffusion process $$X=(X_t)_{t\in [0,1]}$$ X = ( X t ) t ∈ [ 0 , 1 ] observed at discrete times and high frequency, solution of a stochastic differential equation whose drift and diffusion coefficients are assumed to be unknown. In this article, we focus on the nonparametric estimation of the diffusion coefficient. We propose ridge estimators of the square of the diffusion coefficient from discrete observations of X that are obtained by minimization of the least squares contrast. We prove that the estimators are consistent and derive rates of convergence as the number of observations tends to infinity. Two observation schemes are considered in this paper. The first scheme consists in one diffusion path observed at discrete times, where the discretization step of the time interval [0, 1] tends to zero. The second scheme consists in repeated observations of the diffusion process X, where the number of the observed paths tends to infinity. The theoretical results are completed with a numerical study over synthetic data.