Consider a diffusion process X=(X_t), with t in [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 esstimation of the diffusion coefficient. We propose ridge estimators of the square of the diffusion coefficient from discrete observations of X and that are obtained by minimization of the least squares contrast. We prove that the estimators are consistent and derive rates of convergence as the size of the sample paths tends to infinity, and the discretization step of the time interval [0,1] tend to zero. The theoretical results are completed with a numerical study over synthetic data.

翻译：考虑一个在[0,1]区间上的扩散过程X=(X_t)，该过程在离散时间和高频条件下观测，是漂移系数和扩散系数均未知的随机微分方程的解。本文聚焦于扩散系数的非参数估计。我们基于X的离散观测值，通过最小化最小二乘对比度，提出了扩散系数平方的岭估计量。我们证明了这些估计量具有相合性，并推导出当样本路径大小趋于无穷大且时间区间[0,1]的离散化步长趋近于零时的收敛速率。理论结果辅以合成数据的数值研究。