  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.
