Convergence of conditional expectations is considered. The convergence results for conditional expectations are shown to be the same as when the absolutely continuous change of measure can be made directly. A filtering application is examined. If the approximating model uses noise increments with a density which must have compact support, conditions are given which ensure the expected convergence of the conditional expectations.>

For each N, and each fixed time T, a signal X/sup N/ and a 'noisy' observation Y/sup N/ are defined by a pair of stochastic difference equations. Under certain conditions (X/sup N/, Y/sup 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/sup N/) mod Y/sup N/) to E(F(X) mod 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) mod 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/sup N/ which is asymptotically optimal and is an improvement on the usual Kalman-Bucy estimate.>