Gaussian Process Approximations of Stochastic Differential Equations
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Stochastic differential equations arise naturally in a range of contexts, from financial to environmental modelling. Current solution methods are based on a range of strong and weak approximation techniques, however these are limited in their representation of the posterior process in the presence of data. In this work we present a novel Gaussian process approximation to the posterior measure over paths for a general class of stochastic differential equations in the presence of observations. The method is applied to two simple problems: the Ornstein-Uhlenbeck process, of which the exact solution is known and can be compared to, and the double-well system, for which standard approaches such as the ensemble Kalman smoother fail to provide a satisfactory result. Experiments show that our variational approximation is viable and that the results are very promising as the variational approximation outperforms standard Gaussian process regression for non-Gaussian Markov processes.
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