Exact Innnite Dimensional Filters and Explicit Solutions

Previously, we deened innnite dimensional exact lters as nonlinear lters which can be conveniently reduced without approximation to a single convo-lution (plus a simple transformation and substitution). We showed that such problems do exist and the observation process can be far more general than those for exact nite dimensional lters like the Kalman and Benes lters. Moreover, our innnite dimensional exact lters compare favorably in terms of time eeciency and accuracy to other methods except for the nite dimensional exact lters that have limited utility. Herein, we broaden the realm of applicability for our innnite dimensional exact lters including problems with new nonlinear drifts and nonlinear dispersion coeecients. In particular, we investigate the problem of determining which scalar continuous-discrete lter-ing problems can be solved with essentially a single convolution with respect to a standard normal distribution. This leads to a particularly simple ltering algorithm because the Fourier transform of the standard normal distribution is known in closed form and very well behaved.