Основні поняття
A novel non-parametric learning paradigm is proposed for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies on discrete-time observations of the state. The method provides theoretical estimates of non-asymptotic learning rates that become increasingly tighter as the regularity of the unknown coefficients increases.
Анотація
The key aspects of the proposed approach are:
-
Learning the laws of the stochastic differential equation through independent discrete-time observation of the state:
- A RKHS-based model is used to approximate the unknown densities of the state process.
- Theoretical estimates show the approximation error becomes tighter as the number of observations and the regularity of the coefficients increase.
-
Learning finite-dimensional models for the drift and diffusion coefficients by fitting approximated solutions to the Fokker-Planck equation:
- The drift and diffusion coefficients are learned by solving a finite-dimensional convex optimization problem that matches the Fokker-Planck equation evaluated at the RKHS-based density model.
- Theoretical estimates show the error between the learned coefficients and the true ones becomes tighter as the regularity of the coefficients increases.
-
Providing theoretical guarantees for the accuracy of the learned coefficients in the context of observation and regulation of stochastic differential equations:
- By combining the previous results, non-asymptotic learning rates are derived for the identification of the drift and diffusion coefficients.
- These rates become increasingly tighter as the regularity of the coefficients increases.
The proposed method is kernel-based, allowing efficient numerical implementation through offline pre-processing.
Статистика
The following sentences contain key metrics or figures:
The proposed method enjoys non-asymptotic learning rates that become increasingly tighter as the regularity of the unknown drift and diffusion coefficients increases.
The higher the degree of smoothness of the drift and diffusion coefficients, the lower the number of observations and the computational complexity needed to achieve a given precision.
Цитати
"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."