Filtering with Observations on a Riemannian Symmetric Space

Abstract

The point under discussion in this paper refers to a problem of filtering in which observation Y of a system state X is a process with values in a symmetric space M. The observation is a Brownian motion transformed by an isometry of M depending on the state. It takes its values in manifold M and its multiplicative formulation is nonstandard. In many physical situations, e.g., mechanics, robotics, spatial fields, the filtering problems are naturally set up in manifolds as well for the signal and the observation. The reference probability method is used to construct the model. Then filtering equations are deduced; these comply with the conditional law according to its observation. Unique characterization of this conditional law is given. Last, two examples are investigated. First the multivariate case: the observation Y is in $\mathbb{R}^d $ so that $Y_t = \Sigma _t (X)W_t + H_t (X)$, where W is a multivariate Brownian motion; $\Sigma $ (a rotation) and H (a translation) are absolutely continuous with respect to time t. The second example is filtering on the sphere; observation y is of the following form: $Y_t = g_t (X) \cdot W_t $, where W is a Brownian motion on sphere $S_2$ and $g_t (X)$ is a rotation of M absolutely continuous with respect to time t.