On diffusion approximations for filtering
Following the earlier paper by \textit{H. J. Kushner} and \textit{W. J. Runggaldier} [IEEE Trans. Autom. Control AC-32, 123-133 (1987; Zbl 0617.93060)], the authors consider a family of processes \((X^{\epsilon},Y^{\epsilon})\) where \(X^{\epsilon}=(X^{\epsilon}_ t)\) is unobservable, while \(Y^{\epsilon}=(Y^{\epsilon}_ t)\) is observable. The family is given by a model that is nonlinear in the observations, has coefficients that may be rapidly oscillating, and additive disturbances that may be wide- band and non-Gaussian. Using results of diffusion approximation for semimartingales, they show the convergence in distribution (for \(\to 0)\) of \((X^{\epsilon},Y^{\epsilon})\) to a process (X,Y) that satisfies a linear-Gaussian model. Applying the Kalman-Bucy filter for (X,Y) to \((X^{\epsilon},Y^{\epsilon})\), they obtain a linear filter estimate for \((X^{\epsilon}_ t)\), given the observations \(\{Y^{\epsilon}_ s,\quad 0\leq s\leq t\}.\) Such filter estimate is shown to possess the property of asymptotic (for \(\epsilon \to 0)\) optimality of its variance. The results are also applied to show the effects that a limiter in the observation equation may have on the signal-to-noise ratio and thus on the filter variance.
diffusion approximation for semimartingales
scientific article; zbMATH DE number 4211186
scientific article