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The continuous‐time Kalman filter

The Model: Continuous-time linear system, with white noises state and measurement noises (not necessarily Gaussian). Goal: Develop the continuous-time Kalman filter as the optimal linear estimator (L-MMSE) for this system. One way to develop the continuous-time filter is as the limit (with ∆T → 0) of the discrete time case. The derivation below follows a direct approach, based on the innovations process, introduced by Kailath, and will be somewhat informal. A rigorous but very accessible treatment may be found in: M. Davis, Linear Estimation and Stochastic Control, 1977. However, this more advanced treatment is only essential in the nonlinear estimation problem.