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We study Gaussian approximations to the distribution of a diffusion. Theapproximations are easy to compute: they are defined by two simple ordinarydifferential equations for the mean and the covariance. Time correlations canalso be computed via solution of a linear stochastic differential equation. Weshow, using the Kullback-Leibler divergence, that the approximations areaccurate in the small noise regime. An analogous discrete time setting is alsostudied. The results provide both theoretical support for the use of Gaussianprocesses in the approximation of diffusions, and methodological guidance inthe construction of Gaussian approximations in applications.

Gaussian Approximations of Small Noise Diffusions in Kullback-Leibler Divergence