Nonlinear Filtering
Pavel Chigansky (Weizmann Institute)

The first part of this talk is intended as an introduction to the filtering problem for random processes, i.e., the optimal estimation of signals from the past of the their noisy observations. The standard setting here consists of a pair of processes (X,Y)=(X_t,Y_t). where the signal component X is to be estimated at a current time t>0 on the basis of the trajectory of Y , observed up to this t . Under the minimal mean square error (MMSE) criterion, the optimal estimate of X_t is the conditional expectation of X given the process Y up to time t. If both X and (X,Y) are Markov processes, then the conditional distribution satisfies a recursive equation, called filter, which realizes the optimal fusion of the a priori statistical knowledge about the signal and the a posteriori information borne by the observation path.
In the second part, I will touch upon the recent progress in stability problem for nonlinear filters.