Filtering formulae for partially observed linear systems with non-gaussian initial conditions

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Abstract
In this paper, a partially observed linear system is considered with arbitrary non-Gaussian initial conditions and the corresponding (nonlinear) filtering problem is investigated. An explicit formula is obtained for the conditional expectation of an arbitrary function of the current state given past observations; a set of sufficient statistics is shown to exist which are recursively computable as outputs of a finite-dimensional dynamical system. The basic results are specialized to purely complex exponentials and to indicator functions of Bore1 sets, and yield formulae for the conditional characteristic function and probability law of the current state given past observations. A special case when some covariance matrix is invertible, is also studied and a sharpening of the basic results is obtained as the existence and form of a conditional density is established. The method of analysis is probabilistic and relies on Girsanov's Theorem, basic results in linear filtering, theory and some easy facts for Gaussian random variables.

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