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:2015:10.1007/s00498-015-0154-1,
author = {Armstrong, J and Brigo, D},
doi = {10.1007/s00498-015-0154-1},
journal = {Mathematics of Control, Signals, and Systems},
title = {Nonlinear filtering via stochastic PDE projection on mixture manifolds in L^2 direct metric},
url = {http://dx.doi.org/10.1007/s00498-015-0154-1},
volume = {28},
year = {2015}
}

RIS format (EndNote, RefMan)

TY  - JOUR
AB - We examine some differential geometric approaches to finding approximatesolutions to the continuous time nonlinear filtering problem. Our primary focusis a new projection method for the optimal filter infinite-dimensional stochastic partial differential equation (SPDE), based on the direct L2 metric and on a family of normal mixtures. This results in a new finite-dimensional approximate filter based on the differential geometric approach to statistics. We compare this new filter to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and compare the L2 mixture projection filter with a particle method with the same number of parameters, using the Levy metric. We discuss differences between projecting the SPDE for the normalized density, known as Kushner–Stratonovich equation, and the SPDE for the unnormalized density known as Zakai equation. We prove that for a simple choice of the mixture manifold the L2 mixture projection filter coincides with a Galerkin method, whereas for more general mixture manifolds the equivalence doesnot hold and the L2 mixture filter is more general. We study particular systems that may illustrate the advantages of this new filter over other algorithms when comparing outputs with the optimal filter. We finally consider a specific software design that is suited for a numerically efficient implementation of this filter and provide numerical examples. We leverage an algebraic ring structure by proving that in presence of a given structure in the system coefficients the key integrations needed to implement the new filter equations can be executed offline.
AU - Armstrong,J
AU - Brigo,D
DO - 10.1007/s00498-015-0154-1
PY - 2015///
SN - 0932-4194
TI - Nonlinear filtering via stochastic PDE projection on mixture manifolds in L^2 direct metric
T2 - Mathematics of Control, Signals, and Systems
UR - http://dx.doi.org/10.1007/s00498-015-0154-1
UR - http://hdl.handle.net/10044/1/30130
VL - 28
ER -