Robust filtering: Correlated noise and multidimensional observation
In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff and Noordhoff] pointed out that it would be natural for πt, the solution of the stochastic filtering problem, to depend continuously on th...
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| Format: | Journal article |
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Institute of Mathematical Statistics
2013
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| _version_ | 1826271876599513088 |
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| author | Crisan, D Diehl, J Friz, P Oberhauser, H |
| author_facet | Crisan, D Diehl, J Friz, P Oberhauser, H |
| author_sort | Crisan, D |
| collection | OXFORD |
| description | In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff and Noordhoff] pointed out that it would be natural for πt, the solution of the stochastic filtering problem, to depend continuously on the observed data Y={Ys,s∈[0,t]}. Indeed, if the signal and the observation noise are independent one can show that, for any suitably chosen test function f, there exists a continuous map θft, defined on the space of continuous paths C([0,t],Rd) endowed with the uniform convergence topology such that πt(f)=θft(Y), almost surely; see, for example, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff and Noordhoff], Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43–56], Davis [Z. Wahrsch. Verw. Gebiete 54 (1980) 125–139], Davis [Teor. Veroyatn. Primen. 27 (1982) 160–167], Kushner [Stochastics 3 (1979) 75–83]. As shown by Davis and Spathopoulos [SIAM J. Control Optim. 25 (1987) 260–278], Davis [In Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505–528], [In The Oxford Handbook of Nonlinear Filtering (2011) 403–424 Oxford Univ. Press], this type of robust representation is also possible when the signal and the observation noise are correlated, provided the observation process is scalar. For a general correlated noise and multidimensional observations such a representation does not exist. By using the theory of rough paths we provide a solution to this deficiency: the observation process Y is “lifted” to the process Y that consists of Y and its corresponding Lévy area process, and we show that there exists a continuous map θft, defined on a suitably chosen space of Hölder continuous paths such that πt(f)=θft(Y), almost surely. |
| first_indexed | 2024-03-06T22:03:40Z |
| format | Journal article |
| id | oxford-uuid:4f6ea36c-86e2-4065-8cda-d97c8e9feefa |
| institution | University of Oxford |
| last_indexed | 2024-03-06T22:03:40Z |
| publishDate | 2013 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | oxford-uuid:4f6ea36c-86e2-4065-8cda-d97c8e9feefa2022-03-26T16:07:11ZRobust filtering: Correlated noise and multidimensional observationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4f6ea36c-86e2-4065-8cda-d97c8e9feefaSymplectic Elements at OxfordInstitute of Mathematical Statistics2013Crisan, DDiehl, JFriz, POberhauser, HIn the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff and Noordhoff] pointed out that it would be natural for πt, the solution of the stochastic filtering problem, to depend continuously on the observed data Y={Ys,s∈[0,t]}. Indeed, if the signal and the observation noise are independent one can show that, for any suitably chosen test function f, there exists a continuous map θft, defined on the space of continuous paths C([0,t],Rd) endowed with the uniform convergence topology such that πt(f)=θft(Y), almost surely; see, for example, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff and Noordhoff], Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43–56], Davis [Z. Wahrsch. Verw. Gebiete 54 (1980) 125–139], Davis [Teor. Veroyatn. Primen. 27 (1982) 160–167], Kushner [Stochastics 3 (1979) 75–83]. As shown by Davis and Spathopoulos [SIAM J. Control Optim. 25 (1987) 260–278], Davis [In Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505–528], [In The Oxford Handbook of Nonlinear Filtering (2011) 403–424 Oxford Univ. Press], this type of robust representation is also possible when the signal and the observation noise are correlated, provided the observation process is scalar. For a general correlated noise and multidimensional observations such a representation does not exist. By using the theory of rough paths we provide a solution to this deficiency: the observation process Y is “lifted” to the process Y that consists of Y and its corresponding Lévy area process, and we show that there exists a continuous map θft, defined on a suitably chosen space of Hölder continuous paths such that πt(f)=θft(Y), almost surely. |
| spellingShingle | Crisan, D Diehl, J Friz, P Oberhauser, H Robust filtering: Correlated noise and multidimensional observation |
| title | Robust filtering: Correlated noise and multidimensional observation |
| title_full | Robust filtering: Correlated noise and multidimensional observation |
| title_fullStr | Robust filtering: Correlated noise and multidimensional observation |
| title_full_unstemmed | Robust filtering: Correlated noise and multidimensional observation |
| title_short | Robust filtering: Correlated noise and multidimensional observation |
| title_sort | robust filtering correlated noise and multidimensional observation |
| work_keys_str_mv | AT crisand robustfilteringcorrelatednoiseandmultidimensionalobservation AT diehlj robustfilteringcorrelatednoiseandmultidimensionalobservation AT frizp robustfilteringcorrelatednoiseandmultidimensionalobservation AT oberhauserh robustfilteringcorrelatednoiseandmultidimensionalobservation |