As a topic, Stochastic Filtering Theory has progressed rapidly in recent years. For example, the (branching) particle system representation of the optimal filter has been extensively studied to seek more effective numerical approximations of the optimal filter. The stability of the filter with 'incorrect' initial state, as well as the long-term behavior of the optimal filter, has attracted the attention of many researchers, and there are some recent exciting results in singular filtering models. In this text, Jie Xiong introduces the reader to the basics of Stochastic Filtering Theory before covering the key recent advances. The text is written in a clear style suitable for graduates in mathematics and engineering with a background in basic probability.

Preface 1. Introduction 2. Brownian motion and martingales 3. Stochastic integrals and Ito's formula 4. Stochastic differential equations 5. Filtering model and Kallianpur-Striebel formula 6. Uniqueness of the solution for Zakai's equation 7. Uniqueness of the solution for the filtering equation 8. Numerical methods 9. Linear filtering 10. Stability of nonlinear filtering 11. Singular filtering Bibliography Index

Jie Xiong , Department of Mathematics, University of Tennesse

`It is a timely account of the field suitable for serious researchers in stochastic analysis.' Time Higher Education Supplement |d 27/11/2008