Martingale Projection and Linear Filtering in Hilbert Spaces. I: The Theory

Abstract

Using a theorem on the projection of Hilbertian martingales and a Hilbertian theorem of the Girsanov type, previously established by the author, to study the filtering problem for systems governed by linear evolution equations in Hilbert spaces, we show that if the observation is a function of the state through an unbounded operator and with values in a Hilbert space (under some hypotheses on the observation noise), the filtered state is obtained as the solution of a differential stochastic equation, the coefficients of which are given by a solution of an operator differential Ricatti equation. We show that the linear filtering problem can be solved only if the solution of this equation is unique. If the observation is made through a bounded operator, we find again, with the innovation method, results from A. Bensoussan (1971) in the partial differential equation context, and from R. F. Curtain (1975) and S. K. Mitter–R. Vinter (1974) in the linear hereditary equation context; the application to the linear hereditary equation will be presented in another paper.