Communications in Mathematical Sciences

Volume 15 (2017)

Number 7

Gaussian approximations of small noise diffusions in Kullback–Leibler divergence

Pages: 2087 – 2097

(Fast Communication)



Daniel Sanz-Alonso (Mathematics Institute, University of Warwick, Coventry, United Kingdom)

Andrew M. Stuart (Mathematics Institute, University of Warwick, Coventry, United Kingdom)


We study Gaussian approximations to the distribution of a diffusion. The approximations are easy to compute: they are defined by two simple ordinary differential equations for the mean and the covariance. Time correlations can also be computed via solution of a linear stochastic differential equation. We show, using the Kullback–Leibler divergence, that the approximations are accurate in the small noise regime. An analogous discrete time setting is also studied. The results provide both theoretical support for the use of Gaussian processes in the approximation of diffusions, and methodological guidance in the construction of Gaussian approximations in applications.


Gaussian approximations, diffusion processes, small noise, Kullback–Leibler divergence

2010 Mathematics Subject Classification

28C20, 60G15, 60H10, 65L05

Daniel Sanz-Alonso is supported by EPSRC as part of the MASDOC DTC at the University of Warwick with grant No. EP/H023364/1. The work of Andrew M. Stuart is supported by DARPA, EPSRC and ONR.

Received 19 May 2016

Accepted 30 September 3016

Published 16 October 2017