We introduce a nonparametric approach for estimating drift and diffusion functions in systems of stochastic differential equations from observations of the state vector. Gaussian processes are used as flexible models for these functions and estimates are calculated directly from dense data sets using Gaussian process regression. We also develop an approximate expectation maximization algorithm to deal with the unobserved, latent dynamics between sparse observations. The posterior over states is approximated by a piecewise linearized process of the Ornstein-Uhlenbeck type and the maximum a posteriori estimation of the drift is facilitated by a sparse Gaussian process approximation.

We introduce a nonparametric approach for estimating drift functions in systems of stochastic differential equations from incomplete observations of the state vector. Using a Gaussian process prior over the drift as a function of the state vector, we develop an approximate EM algorithm to deal with the unobserved, latent dynamics between observations. The posterior over states is approximated by a piecewise linearized process and the MAP estimation of the drift is facilitated by a sparse Gaussian process regression.