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

TL;DR

This paper introduces two new optimal projection methods for approximating high-dimensional SDEs on submanifolds, improving existing techniques with rigorous optimality criteria and demonstrating their effectiveness in filtering applications.

Contribution

The paper develops the Ito-vector and Ito-jet projections, providing a systematic, optimal approximation framework for SDEs on submanifolds that generalizes and improves upon previous methods.

Findings

Ito projections are optimal in the mean-square sense.

The approach outperforms classical filters like the Extended Kalman Filter.

Numerical results confirm the efficacy of the new projections.

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. We also show that 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 and 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 distribution. We perform a numerical comparison of our optimally approximated filter with the classical Extended Kalman Filter to demonstrate the efficacy of the approach.