Estimation of drift and diffusion functions from time series data: A maximum likelihood framework

Complex systems are characterized by a huge number of degrees of freedom often interacting in a nonlinear manner. In many cases macroscopic states, however, can be characterized by a small number of order parameters that obey stochastic dynamics in time. Recently, techniques for the estimation of the corresponding stochastic differential equations from measured data have been introduced. This paper develops a framework for the estimation of the functions and their respective (Bayesian posterior) confidence regions based on likelihood estimators. In succession, approximations are introduced that significantly improve the efficiency of the estimation procedure. While being consistent with standard approaches to the problem, this paper solves important problems concerning the applicability and the accuracy of estimated parameters.