Projection smoothing for continuous and continuous-discrete stochastic dynamic systems

This study investigates the states of continuous and continuous-discrete time nonlinear stochastic dynamic systems conditioned on noisy measurements. We adopt a differential geometric approach to construct finite-dimensional algorithms for solving the filtering and smoothing problems associated with such systems. In particular, we use a projection method based on the Hellinger distance and the related Fisher metric to derive a novel backward equation that is satisfied by the approximate probability density associated with the smoothing problem. Finally, by combining our approach with a previously developed projection filter, we formulate a finite-dimensional approximation of the forward (filtering) and backward (smoothing) algorithms on the basis of the above-mentioned projection method.