EXPLICIT FILTERS FOR DIFFUSIONS WITH CERTAIN NONLINEAR DRIFTS.

LET X(T) BE A DIFFUSION SATISFYING THE STOCHASTIC DIFFERENTIAL EQUATION DX(T) = F(X(T))DT + DB(T), WHERE F ' (X) = F**2(X) = AX**2 + BX + C, A > 0. V. BENES GAVE AN EXPLICIT FORMULA FOR THE CONDITIONAL DENSITY OF X(T) GIVEN Y(S), 0 < S < T, WHERE Y(S) = ″INTEGRAL″ **T//0X(S)DS + W(T), WHEN W( * ) IS A BROWNIAN PROCESS INDEPENDENT OF X( * ). THE RESULT IS EXTENDED AND THEN APPLIED TO DERIVE RECURSIVE FILTERING EQUATIONS FOR ESTIMATING CONDITIONAL MOMENTS, FOR ESTIMATING POLYNOMIAL FUNCTIONALS OF X( * ), AND FOR SMOOTHING.