Projections of SDEs onto submanifolds
In Armstrong et al. (Proc Lond Math Soc (3) 119(1):176–213, 2019) the authors define three projections of ℝ𝑑-valued stochastic differential equations (SDEs) onto submanifolds: the Stratonovich, Itô-vector and Itô-jet projections. In this paper, after a brief survey of SDEs on manifolds, we begin by...
| Main Authors: | , , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Springer
2023
|
| _version_ | 1826310480006742016 |
|---|---|
| author | Armstrong, J Brigo, D Ferrucci, E |
| author_facet | Armstrong, J Brigo, D Ferrucci, E |
| author_sort | Armstrong, J |
| collection | OXFORD |
| description | In Armstrong et al. (Proc Lond Math Soc (3) 119(1):176–213, 2019) the authors define three projections of ℝ𝑑-valued stochastic differential equations (SDEs) onto submanifolds: the Stratonovich, Itô-vector and Itô-jet projections. In this paper, after a brief survey of SDEs on manifolds, we begin by giving these projections a natural, coordinate-free description, each in terms of a specific representation of manifold-valued SDEs. We proceed by deriving formulae for the three projections in ambient ℝ𝑑-coordinates. We use these to show that the Itô-vector and Itô-jet projections satisfy respectively a weak and mean-square optimality criterion “for small t”: this is achieved by solving constrained optimisation problems. These results confirm, but do not rely on the approach taken in Armstrong et al. (Proc Lond Math Soc (3) 119(1):176–213, 2019), which is formulated in terms of weak and strong Itô–Taylor expansions. In the final section we exhibit examples showing how the three projections can differ, and explore alternative notions of optimality. |
| first_indexed | 2024-03-07T07:52:35Z |
| format | Journal article |
| id | oxford-uuid:2ae8a16c-67d4-470b-8cb5-a31a433552c9 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:52:35Z |
| publishDate | 2023 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:2ae8a16c-67d4-470b-8cb5-a31a433552c92023-08-01T12:36:47ZProjections of SDEs onto submanifoldsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2ae8a16c-67d4-470b-8cb5-a31a433552c9EnglishSymplectic ElementsSpringer2023Armstrong, JBrigo, DFerrucci, EIn Armstrong et al. (Proc Lond Math Soc (3) 119(1):176–213, 2019) the authors define three projections of ℝ𝑑-valued stochastic differential equations (SDEs) onto submanifolds: the Stratonovich, Itô-vector and Itô-jet projections. In this paper, after a brief survey of SDEs on manifolds, we begin by giving these projections a natural, coordinate-free description, each in terms of a specific representation of manifold-valued SDEs. We proceed by deriving formulae for the three projections in ambient ℝ𝑑-coordinates. We use these to show that the Itô-vector and Itô-jet projections satisfy respectively a weak and mean-square optimality criterion “for small t”: this is achieved by solving constrained optimisation problems. These results confirm, but do not rely on the approach taken in Armstrong et al. (Proc Lond Math Soc (3) 119(1):176–213, 2019), which is formulated in terms of weak and strong Itô–Taylor expansions. In the final section we exhibit examples showing how the three projections can differ, and explore alternative notions of optimality. |
| spellingShingle | Armstrong, J Brigo, D Ferrucci, E Projections of SDEs onto submanifolds |
| title | Projections of SDEs onto submanifolds |
| title_full | Projections of SDEs onto submanifolds |
| title_fullStr | Projections of SDEs onto submanifolds |
| title_full_unstemmed | Projections of SDEs onto submanifolds |
| title_short | Projections of SDEs onto submanifolds |
| title_sort | projections of sdes onto submanifolds |
| work_keys_str_mv | AT armstrongj projectionsofsdesontosubmanifolds AT brigod projectionsofsdesontosubmanifolds AT ferruccie projectionsofsdesontosubmanifolds |