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 It'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 XiongDepartment of Mathematics, University of Tennesse