Topics in stochastic nonlinear filtering

The primary goal of nonlinear filtering theory is to obtain information about the signal observed in the presence of an additive noise. This is done by studying the conditional distribution of the unobservable signal {dollar}Xsb{lcub}t{rcub}{dollar} given the observations Y up to the time instant 't', i.e., {dollar}sigma{lcub}Ysb{lcub}s{rcub}, s in lbrack 0,trbrack{rcub}.{dollar}; A Bayes type formula, extending the one by Kallianpur-Striebel, for the conditional expectation of {dollar}g(Xsb{lcub}t{rcub}){dollar} given {dollar}sigma{lcub}Ysb{lcub}s{rcub}, s in lbrack 0, trbrack{rcub}{dollar} is obtained when the noise process is a general Gaussian process. The result is then applied to the special case where the noise process is equivalent to the Wiener process.; Another generalization of the usual nonlinear filtering theory when the coefficient function 'h' appearing in the filtering model depends not only on the instantaneous value of the signal {dollar}Xsb{lcub}t{rcub}{dollar}, but also on the past signal values, is studied. The signal process is modeled by a stochastic delay differential equation (SDDE) which, unlike the usual filtering theory, makes it a non-Markov process. The signal process is characterized as the unique solution to an appropriate martingale problem. A Zakai type stochastic differential equation (SDE) is obtained for the optimal filter corresponding to the nonlinear filtering problem and the filter is characterized as the unique solution to the Zakai equation.