Nonlinear Filtering Formulas for Discrete-Time Observations
Yoshiki Takeuchi, Hajime AKASHI · SIAM Journal on Control and Optimization · 1981
This paper presents two types of nonlinear filtering formulas in the form of differential equations for the case where the signal is a continuous-time and the observation is a discrete-time process. The observation is corrupted by additive Gaussian white noise. The method of solution is based on Girsanov’s measure transformation technique and a family of probability measures is introduced which is indexed by the continuous-time parameter. By computing the time evolution of these measures, the conditional expectation of a functional of the signal, given the observations, with respect to the original measure is smoothly updated. The obtained formulas are recursive with respect to the observation sequence whereas the well-known Bayes’ formula is nonrecursive in the general case considered.