8. Continuous Time: Filtering Algorithms

In this chapter, we describe various algorithms for determination of the filtering distribution μ ^t in continuous time. We begin in Section 8.1 with the Kalman–Bucy filter, which provides an exact algorithm for linear problems. Since the filtering distribution is Gaussian in this case, the distribution is entirely characterized by the mean and covariance; the algorithm comprises a system of differential equations for the mean and the covariance. In Section 8.2, we discuss the approximate Gaussian methods introduced in Section 4.​2 in the discrete-time setting. Similarly to the case of the Kalman–Bucy filter, we again obtain a differential equation for the mean; for the extended Kalman (ExKF) filter, we also obtain an equation for the covariance, and for the ensemble Kalman filter (EnKF), we have a system of differential equations coupled through their empirical mean and covariance. In Section 8.3, we discuss how the particle filter methodology introduced in Section 4.​3 extends to the continuous case, while in Section 8.4, we study the long-time behavior of some of the filtering algorithms discussed in the previous sections. Finally, in Section 8.5, we present some numerical illustrations and conclude with bibliographic notes and exercises in Sections 8.6 and 8.7 respectively.