We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker-Planck equation to such observations, yielding theoretical estimates of non-asymptotic learning rates which, unlike previous works, become increasingly tighter when the regularity of the unknown drift and diffusion coefficients becomes higher. Our method being kernel-based, offline pre-processing may be profitably leveraged to enable efficient numerical implementation, offering excellent balance between precision and computational complexity.

翻译：我们提出了一种新颖的非参数学习范式，用于识别多维非线性随机微分方程的漂移和扩散系数，该方法依赖于状态的高散时间观测。核心思想本质上是将基于再生核希尔伯特空间近似的福克-普朗克方程拟合到这些观测数据，从而得到非渐近学习率的理论估计——与以往研究不同，当未知漂移和扩散系数的正则性提高时，这些估计会变得更紧缩。由于我们的方法基于核方法，可以充分利用离线预处理实现高效数值计算，在精度与计算复杂度之间达到卓越的平衡。