The time-continuous filtering problem leads to the Kushner-Zaikai equation at the level of conditional probabilities. More recently it has been shown that an analog formulation at the level of stochastic equations leads to feedback control laws involving the data and a Kalman-like gain function. Such formulations are of interest for several reasons. In my talk, I will focus on the following three aspects:
i) the development of robust filtering algorithms and uncertainty quantification,
ii) connection between data assimilation and optimal control problems,
iii) theoretical analysis of filter algorithms as interacting particle systems.