In this thesis a certain estimation problem for nonlinear stochastic dynamical systems in discrete time, known as filtering in the literature, is considered. The objective is to reconstruct the current state of the system by means of observations. The observations are noise corrupted measurements of a function of the state.

In the first chapter we will provide concepts from probability theory necessary to define and investigate the nonlinear filtering problem. The definition of the nonlinear filter is presented in the second section of this chapter.

The second chapter explains in detail why filtering is an extremely difficult problem in a nonlinear context. It turns out that a finite dimensional representation of the filter is possible in very special circumstances, only. This chapter summarizes known results and is intended mainly as a motivation for the necessity of investigating approximation schemes for the nonlinear filter, considered in the following two chapters.

Numerical approximation methods are then investigated from a general point of view in the third chapter. A general framework to obtain error bounds for approximation schemes is presented. Necessary for this investigation are metrics for probability distributions we will make extensive use of. Furthermore, an essential property of the filter required to give a bounded approximation error turns out to be a negative Lyapunov exponent of the filter dynamics.

The fourth chapter provides some classes of approximation methods. Common to all these methods are projection techniques on parametrized families of probability distributions. This approach was carried out for continuous time systems already by several authors. However, the error analysis carried out in this thesis is, to the best of our knowledge, new.

The last two chapters present (aside from two small sections devoted to a Monte Carlo approach), two interesting and important applications of nonlinear filtering. The first is estimation of an unknown parameter in the dynamics. This problem is very important in all branches of science, and we will present two numerical examples. The second application is reconstruction of a sent message in telecommunications. To this problem a chapter is devoted, including results on the bit error probability obtained using methods from nonlinear filtering theory for a simple transmitter model.

To summarize, the aim of this thesis is threefold. First, to show that filtering of nonlinear dynamical system is a nontrivial and interesting problem from a mathematical point of view, second to show methods to overcome the difficulties arising in applications, and third to show that filtering is not a purely artificial mathematical problem but has a great significance in science and engineering.