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Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections

Research output: Contribution to journalArticle

John Armstrong, Damiano Brigo, Emilio Rossi Ferrucci

Original languageEnglish
Pages (from-to)176-213
Number of pages38
JournalPROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
Volume119
Issue number1
Early online date21 Dec 2018
DOIs
Accepted/In press11 Dec 2018
E-pub ahead of print21 Dec 2018
PublishedJul 2019

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Abstract

We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Ito-vector and Ito-jet projections. This allows one to systematically develop low dimensional approximations to high dimensional SDEs using differential geometric techniques. The approach generalizes the notion of projecting a vector field onto a submanifold in order to derive approximations to ordinary differential equations, and improves the previous Stratonovich projection method by adding optimality analysis and results. Indeed, just as in the case of ordinary projection, our definitions of projection are based on optimality arguments and give in a well-defined sense "optimal" approximations to the original SDE in the mean-square sense over small times. We also explain how the Stratonovich projection satisfies an optimality criterion that is more ad hoc and less appealing than the criteria satisfied by the Ito projections we introduce.

As an application, we consider approximating the solution of the non-linear filtering problem with a Gaussian distribution. We show how the newly introduced Ito projections lead to optimal approximations in the Gaussian family and briefly discuss the optimal approximation for more general families of distributions. We perform a numerical comparison of our optimally approximated filter with the classical Extended Kalman Filter to demonstrate the efficacy of the approach.

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