Robust Filtering: Correlated Noise and Multidimensional Observation

Details Robust Filtering: Correlated Noise and Multidimensional Observation Robust Filtering: Correlated Noise and Multidimensional Observation

2012-01-09

arxiv.org/abs/1201.1858v1

Mathematics

Probability

Scientific paper

In the late seventies, Clark [The design of robust approximations to the stochastic differential equations of nonlinear filtering, Communication systems and random process theory, 1978] 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 $C([0,t],\R^d)$ endowed with the uniform convergence topology such that $\pi_t(f)=\theta^f_{t}(Y)$, almost surely. As shown by Davis [Pathwise nonlinear filtering, Stochastic systems: the mathematics of filtering and identification and applications, 1981] this type of \emph{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 L\'evy area process and we show that there exists a continuous map $\theta_{t}^f$, defined on a suitably chosen space of H\"older continuous paths such that $\pi_t(f)=\theta_{t}^f(\mathbf{Y})$, almost surely.

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