We study efficiency of non-parametric estimation of diffusions (stochastic differential equations driven by Brownian motion) from long stationary trajectories. First, we introduce estimators based on conditional expectation which is motivated by the definition of drift and diffusion coefficients. These estimators involve time- and space-discretization parameters for computing expected values from discretely-sampled stationary data. Next, we analyze consistency and mean squared error of these estimators depending on computational parameters. We derive relationships between the number of observational points, time- and space-discretization parameters in order to achieve the optimal speed of convergence and minimize computational complexity. We illustrate our approach with numerical simulations.

翻译：我们研究从长期固定轨道对散射(由布朗运动驱动的随机差分方程)进行非参数估计的效率。 首先,我们引入基于根据漂移和散射系数定义所驱动的有条件期望的测算器。这些测算器涉及从离散抽样的固定数据中计算预期值的时间和空间分解参数。 其次,我们根据计算参数分析这些测算器的一致性和平均平方差差差差。我们从观测点、时间和空间分解参数的数目中得出各种关系,以便达到最佳的趋同速度并尽量减少计算的复杂性。我们用数字模拟来说明我们的做法。