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Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections
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Title: | Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections |
Authors: | Armstrong, J Brigo, D Rossi Ferrucci, E |
Item Type: | Journal Article |
Abstract: | We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Itô‐vector and Itô‐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 Itô 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 Itô 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. |
Issue Date: | 1-Jul-2019 |
Date of Acceptance: | 11-Dec-2018 |
URI: | http://hdl.handle.net/10044/1/66775 |
DOI: | https://dx.doi.org/10.1112/plms.12226 |
ISSN: | 1460-244X |
Publisher: | London Mathematical Society |
Start Page: | 176 |
End Page: | 213 |
Journal / Book Title: | Proceedings of the London Mathematical Society |
Volume: | 119 |
Issue: | 1 |
Copyright Statement: | © 2018 London Mathematical Society |
Keywords: | Science & Technology Physical Sciences Mathematics 58A20 39A50 58J65 60H10 60J60 (primary) 65D18 (secondary) math.PR math.PR math.DG 58A20, 39A50, 58J65, 60H10, 60J60, 65D18 0101 Pure Mathematics 0104 Statistics |
Publication Status: | Published |
Online Publication Date: | 2018-12-21 |
Appears in Collections: | Financial Mathematics Mathematics Faculty of Natural Sciences |