Non-Parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence
This provides a method for learning stochastic differential equations with improved theoretical guarantees, which is incremental but addresses a known bottleneck in non-parametric estimation.
The paper tackles the problem of identifying drift and diffusion coefficients in multi-dimensional non-linear stochastic differential equations from discrete-time observations, achieving non-asymptotic learning rates that tighten with higher regularity of the unknown coefficients.
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.