Stochastic Differential Systems Filtering * Analysis and

THIS BOOK is concerned with dynamical systems described by stochastic differential equations. The first part of the book deals with the analysis of such systems and the remainder of the book is on application to the filtering problem. The stochastic term driving the dynamical system is taken to be an arbitrary process with independent increments, so that the usual Wiener process model is considered as a special case. There are excellent mathematical texts on filtering for stochastic differential systems (see, for example, Lipster and Shiryayev (1977), and Kallianpur (1980)). The book under review does not discuss filtering in that generality and stresses the engineering aspect throughout. In fact, the non-linear filtering problem is not solved in this book and only approximate non-linear filtering algorithms are developed. The present book is a combination of stochastic signal analysis and continuous-time filtering theory. There are, of course, numerous engineering texts on stochastic signals and systems (some recent publications include those of Gray and Davisson (1986), Mortensen (1987), and Stark and Woods (1986); see also Wong and Hajek (1985) for a more advanced treatment). There are also standard "engineering" books on filtering theory where continuoustime models are handled in detail (Maybeck, 1982; Jazwinski, 1970). None of these books cover both these topics in as much detail as the book under review. Chapter 1 starts with preparatory material on deterministic dynamical systems, including concepts such as inputs, outputs and states of a system, characteristics of linear systems and results on linear differential equations. The chapter ends with an intuitive introduction to stochastic differential systems as extensions of deterministic differential systems with stochastic forcing terms. Chapter 2 gives an introduction to stochastic processes. It starts with the definition of a stochastic process and its construction based on a family of finite-dimensional distributions. The second-order moment leads to covariance and cross-covariance functions and, in special cases, to white noise and Gaussian processes. This is followed by a section on orthogonal expansion of finite-dimensional densities of a stochastic process. This includes Hermite polynomial expansion, Edgeworth series and related orthogonal expansions. This section contains material not usually found on this topic. These expansions play an important role later in the book dealing with the filtering problem. The chapter ends with the mean-square calculus of stochastic processes. Chapter 3 deals with stochastic integrals and stochastic differential equations. Stochastic integrals are first defined for non-random integrands with respect to an arbitrary process with uncorrelated increments. In this connection white noise is introduced as the derivative of a process with an uncorrelated increment. Stochastic measures are defined in order to define stochastic integrals for non-random functions of vector arguments. The authors then study linear stochastic differential equations. Stochastic integrals for