A Semi-Analytical Approach to Solving the Ornstein--Uhlenbeck Equation via Reproducing Kernel Hilbert Spaces (RKHS)

Abstract

This work proposes a semi-analytical method to approximate the solution of the Ornstein--Uhlenbeck (OU) stochastic differential equation by means of Reproducing Kernel Hilbert Spaces (RKHS). The OU process, widely used as a Gaussian mean-reverting model in physics and finance, is decomposed into a deterministic mean-reversion component and a purely stochastic component represented as an It\^o integral. The deterministic part is reconstructed via regularized kernel regression, whereas the stochastic term is modeled through an autoregressive representation in an RKHS induced by a positive definite kernel. By combining the representer theorem with a kernel-based autoregressive scheme, we obtain a finite-dimensional linear system for the kernel coefficients and a fixed-point algorithm for solving the preimage problem. The proposed framework yields a fully kernel-based approximation of the OU dynamics. The methodology is validated on synthetic Ornstein--Uhlenbeck trajectories and, subsequently, on real time series data, illustrating that the RKHS-based approach accurately captures the mean-reversion behavior and stochastic variability of the process while preserving numerical stability and interpretability.