Imperial College London

ProfessorDamianoBrigo

Faculty of Natural SciencesDepartment of Mathematics

Chair in Mathematical Finance
 
 
 
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Contact

 

damiano.brigo

 
 
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Location

 

805Weeks BuildingSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@article{Armstrong:2018:10.1112/plms.12226,
author = {Armstrong, J and Brigo, D and Rossi, Ferrucci E},
doi = {10.1112/plms.12226},
journal = {Proceedings of the London Mathematical Society},
title = {Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections},
url = {http://dx.doi.org/10.1112/plms.12226},
year = {2018}
}

RIS format (EndNote, RefMan)

TY  - JOUR
AB - We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Itôvector and Itôjet projections. This allows one to systematically develop lowdimensional approximations to highdimensional 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 welldefined sense ‘optimal’ approximations to the original SDE in the meansquare 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 nonlinear 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.
AU - Armstrong,J
AU - Brigo,D
AU - Rossi,Ferrucci E
DO - 10.1112/plms.12226
PY - 2018///
SN - 1460-244X
TI - Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections
T2 - Proceedings of the London Mathematical Society
UR - http://dx.doi.org/10.1112/plms.12226
UR - http://hdl.handle.net/10044/1/66775
ER -