Convergence of filters with applications to the Kalman-Bucy case

For each N, and each fixed time T, a signal X^N and a `noisy´ observation Y^N are defined by a pair of stochastic difference equations. Under certain conditions (X^N, Y^N) converges in distribution to (X, Y, where dX(t)= f(t, X(t))dt+dV( t), dY(t)=g(t, X( t))dt+dW(t). Conditions are found under which convergence in distribution of the conditional expectations E{F(X^N)|Y^N} to E{F(X)|Y} follows, for every bounded continuous function F. The case in which the conditional expectations still converge but the limit is not E{ F(X)|Y} is also studied. In the situation where f and g are linear functions of X, an examination of this limit leads to a Kalman-Bucy-type estimate of X ^N which is asymptotically optimal and is an improvement on the usual Kalman-Bucy estimate