Approximate Gaussian process inference for the drift of stochastic differential equations

We introduce a nonparametric approach for estimating drift functions in systems of stochastic differential equations from sparse 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 of the Ornstein-Uhlenbeck type and the MAP estimation of the drift is facilitated by a sparse Gaussian process regression.