On exact filters for continuous signals with discrete observations

Many filtering applications have continuous state dynamics X_t =∫₀^tm(X_s)ds+σW_t +ρ, discrete observations Y_j=Y(t_j), and nonadditive or non-Gaussian observation noise. One wants to calculate the conditional probability Pr{Xt∈dz|Y_j, 0⩽t_j ⩽t} economically. In this paper we show that a combination of convolution, scaling, and substitutions efficiently solves this problem under certain conditions. Our method is easy to use and assumes nothing about the observations other than the ability to construct p(Y_j )|X(t_j), the conditional density of the jth observation given the current state