Results on the filtering problem for linear systems with non-Gaussian initial conditions

The filtering problem for partially observed linear systems in additive Gaussian white noise is studied when the initial state condition has arbitrary, thus a priori non-Gaussian, statistics. The conditional probability law of the current state given past observations is shown to admit a set of sufficient statistics which are recursively computable as outputs of a finite-dimensional system. These results are read off an explicit expression for the conditional characteristic function, obtained with no assumption on the moments or the absolute continuity of the initial state distribution. The methodology behind the computation is briefly outline: the derivation is probabilistic and relies on an absolutely continuous change of measure combined to standard results of linear filtering theory.