Infinite-Dimensional Filtering: The Kalman-Bucy Filter in Hilbert Space

We examine the question of determining the "best" linear filter, in an expected squared error sense, for a signal generated by stochastic linear differential equation on a Hilbert space. Our results, which extend the development in Kalman and Bucy (1960), rely heavily on the integration theory for Banach-space-valued functions of Dunford and Schwartz (1958). In order to derive the Kalman-Bucy filter, we also need to define and discuss such concepts as stochastic process, covariance, orthogonal increments, Wiener process, and stochastic integral in a Hilbert space context. We do this making extensive use of the ideas in Doob (1953). The two crucial points in our treatment are (1) our definition of the covariance as a bounded linear transformation, and (2) our use of a Fubini4ype theorem involving the interchange of stochastic and Lebesgue integration. As a byproduct, we also obtain a fully rigorous theory for the finite-dimensional case which does not rely on Ito's Lemma (cf. Kushner, 1964). This is of some independent interest. The remainder of the paper is divided into the following sections: 2. Preliminaries; 3. Wiener Processes; 4. Stochastic Integration; 5. An Existence Theorem; 6. The Wiener-Hopf Equation; 7. The Optimal Filter; 8. Concluding Remarks. We introduce some basic preliminary notions, the mos~ important of which is the eovariance of two Hilbert-space-valued random variables, in Section 2. Then, we discuss Wiener processes and construct an infinite-