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October 2013 Robust filtering: Correlated noise and multidimensional observation
D. Crisan, J. Diehl, P. K. Friz, H. Oberhauser
Ann. Appl. Probab. 23(5): 2139-2160 (October 2013). DOI: 10.1214/12-AAP896

Abstract

In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff & Noordhoff] pointed out that it would be natural for $\pi_{t}$, the solution of the stochastic filtering problem, to depend continuously on the observed data $Y=\{Y_{s},s\in[0,t]\}$. 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 $\theta^{f}_{t}$, defined on the space of continuous paths $C([0,t],\mathbb{R} ^{d})$ endowed with the uniform convergence topology such that $\pi_{t}(f)=\theta^{f}_{t}(Y)$, 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 & 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 $\mathbf{Y}$ that consists of $Y$ and its corresponding Lévy area process, and we show that there exists a continuous map $\theta_{t}^{f}$, defined on a suitably chosen space of Hölder continuous paths such that $\pi_{t}(f)=\theta_{t}^{f}(\mathbf{Y})$, almost surely.

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D. Crisan. J. Diehl. P. K. Friz. H. Oberhauser. "Robust filtering: Correlated noise and multidimensional observation." Ann. Appl. Probab. 23 (5) 2139 - 2160, October 2013. https://doi.org/10.1214/12-AAP896

Information

Published: October 2013
First available in Project Euclid: 28 August 2013

zbMATH: 1296.60097
MathSciNet: MR3134732
Digital Object Identifier: 10.1214/12-AAP896

Subjects:
Primary: 60G35, 93E11

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.23 • No. 5 • October 2013
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