Approximate Bayes learning of stochastic differential equations
This work addresses the challenge of learning stochastic differential equations from data, which is important for researchers in fields like physics and finance, but it appears incremental as it builds on existing Gaussian process and expectation maximization techniques.
The paper tackles the problem of estimating drift and diffusion functions in stochastic differential equations from state observations, using Gaussian processes for flexible modeling and an approximate expectation maximization algorithm to handle sparse data, achieving direct estimation from dense datasets and efficient computation with sparse approximations.
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.