Optimal Projection Filters

TL;DR

This paper introduces two new optimal projection methods for approximating high-dimensional stochastic differential equations on submanifolds, improving the systematic development of low-dimensional models with applications to nonlinear filtering.

Contribution

It develops the Ito-vector and Ito-jet projections based on optimality principles, providing a more natural and mathematically rigorous approach than previous ad hoc methods.

Findings

The new projections are optimal in the mean-square sense.

They outperform the Stratonovich projection in approximation quality.

Application to nonlinear filtering yields the optimal projection filters.

Abstract

We present the two new notions of projection of a stochastic differential equation (SDE) onto a submanifold, as developed in Armstrong, Brigo e Rossi Ferrucci (2019, 2018): the Ito-vector and Ito-jet projections. This allows one to systematically and optimally develop low dimensional approximations to high dimensional SDEs using differential geometric techniques. Our new projections are based on optimality arguments and yield a well-defined ``optimal'' approximation to the original SDE in the mean-square sense. We also show that the earlier Stratonovich projection satisfies an optimality criterion that is more ad hoc and less natural than the criteria satisfied by the new projections. As an application, we consider approximating the solution of the non-linear filtering problem within a given manifold of densities, using either the Hellinger or $L^2$ direct metrics and related Information Geometry structures on the space of densities. The Stratonovich projection had yielded the projection filters studied in Brigo, Hanzon and Le Gland (1998, 1999), while the new projections lead to the optimal projection filters. The optimal projection filters have been introduced in Armstrong, Brigo e Rossi Ferrucci (2019), where numerical examples for the Gaussian case are given and where they are compared to more traditional nonlinear filters.