Stochastic filtering theory and its stability aspects for deterministic signals

Abstract :  In practice, the state of many systems of interest is accessible only through indirect noisy observations. Stochastic filtering theory deals with estimating the state of the system at a particular instant given some noisy observations of the system up to that instant. It gives the best estimate in the sense of mean square. As it turns out, the state estimation depends on the initial condition of the system, in addition to the observations. Since, in practice, we may not have the knowledge of the initial condition, it is desirable to have the state estimator (filter) behave asymptotically independent of the initial condition. This property is referred to as filter stability. In this talk, we will introduce the formulation of stochastic filtering theory (in the common setting of diffusions) and the problem of filter stability. We present the results of stability for systems with deterministic signals (which is often the case in geosciences).  It will be shown that if the observations are rich enough and the dynamics is chaotic enough then the filter is stable. Since the system is deterministic, we will see that the long time behavior of the filter depends on the characteristics of the dynamics such as the attractor.