On diffusion approximations for filtering

We consider a family of processes (X[var epsilon], Y[var epsilon]) where X[var epsilon] = (X[var epsilon]t) is unobservable, while Y[var epsilon] = (Y[var 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, we show the convergence in distribution (for [var epsilon] --> 0) of (X[var epsilon], Y[var epsilon]) to a process (X, Y) that satisfies a linear-Gaussian model. Applying the Kalman-Bucy filter for (X, Y) to (X[var epsilon], Y[var epsilon]), we obtain a linear filter estimate for X[var epsilon]t, given the observations {Y[var epsilon]s, 0 [less-than-or-equals, slant] s [less-than-or-equals, slant] t}. Such filter estimate is shown to possess the property of asymptotic (for [var epsilon] --> 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.