Abstract
i. We study the problem of the optimal estimation of an infinite-dimensional diffusion process, given a finite-dimensional vector of observation. For finite-dimensional diffusion processes, an analogous filtering problem can be solved with the help of the known KalmanBusy equations [i]. We derive equations for the covariance function and the best leastsquare estimator of an infinite-dimensional diffusion process, described by a stochastic parabolic partial differential equation. The obtained equations are natural analogs of Kalman-Busy equations and are applicable in the filtering problem for infinite-dimensional diffusion processes.
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