Exact finite dimensional nonlinear filters for continuous time processes with discrete time measurements

An exact finite dimensional filter is derived for random processes with certain nonlinear dynamics, that evolve continuously in time and which are observed at discrete points in time with linear measurements corrupted by additive white Gaussian noise. The nonlinear continuous time dynamics must satisfy two conditions that are nearly identical to those recently used by V. E. Benes to derive exact finite dimensional filters for continuous time dynamics and continuous time measurements. As usual, the mathematical tools required to deal with discrete time measurements are much simpler than for continuous time measurements, which makes the discrete time theory accessible to a wider audience. Furthermore, the computational requirements to implement the new discrete time filter are comparable to the Kalman filter. A number of simple approximation techniques are suggested for practical applications in which the dynamics do not satisfy the conditions used by Benes. These approximations are analogous to the so-called "extended Kalman filter," and they represent a generalization of the standard linearization method.