## The Annals of Applied Probability

### Robust filtering: Correlated noise and multidimensional observation

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 5 (2013), 2139-2160.

Dates
First available in Project Euclid: 28 August 2013

https://projecteuclid.org/euclid.aoap/1377696308

Digital Object Identifier
doi:10.1214/12-AAP896

Mathematical Reviews number (MathSciNet)
MR3134732

Zentralblatt MATH identifier
1296.60097

#### Citation

Crisan, D.; Diehl, J.; Friz, P. K.; Oberhauser, H. Robust filtering: Correlated noise and multidimensional observation. Ann. Appl. Probab. 23 (2013), no. 5, 2139--2160. doi:10.1214/12-AAP896. https://projecteuclid.org/euclid.aoap/1377696308

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