Semiparametric maximum likelihood reconstruction of stochastic differential equations driven by white and correlated noise

A semiparametric methodology for reconstructing Markovian and non-Markovian Langevin equations from time series data using unscented Kalman filtering is introduced and explored. The drift function and the logarithm of the diffusion function are expanded into sets of polynomial basis functions. In contrast to the more common state augmentation approach, the Kalman filter is here used only for state estimation and propagation; the model parameters are determined by maximum likelihood based on the predictive distribution generated by the Kalman filter. Model selection regarding the number of included drift and diffusion basis functions is performed with the Bayesian information criterion. The method is successfully validated on various simulated datasets with known system dynamics; it achieves accurate identification of drift and diffusion functions, also outside the prescribed model class, from datasets of moderate length with medium computational cost.