Identifying stochastic differential equation with high-order accuracy via Fourier series

This paper extends the current theory of stochastic differential equation identification by introducing high-order derivatives of the drift term. The proposed Fourier series–based identification technique ensures that the order of neglected terms in the identification is higher than the square of the time step, enabling accurate identification from data collected with relatively large time steps. By approximating the drift term with a Fourier series, the method eliminates the need for prior knowledge or manual basis function library construction. Numerical examples demonstrate that the identified systems accurately capture transient and stationary probability density functions, as well as the stochastic P-bifurcations.