Xiong J. An Introduction to Stochastic Filtering Theory

Oxford University Press, 2008. — 285 p.The object of stochastic filtering is to use probability tools to estimate unobservable stochastic processes that arise in many applied fields including communication, target tracking, and mathematical finance.
Stochastic filtering theory has seen a rapid development in recent years. First, the (branching) particle-system representation of the optimal filter has been studied by many authors to seek more effective numerical approximations of the optimal filter. It turns out that such a representation can be utilized to prove the uniqueness of the solution to the filtering equation itself and, hence, broadening the scope of the tractable class of models.
Secondly, the stability of the filter with “incorrect” initial state as well as the long-time behavior of the optimal filter has attracted the attention of many researchers. This direction of research has become extremely challenging after a gap in a widely cited paper was discovered.
Finally, many problems in mathematical finance, for example, the stochastic volatility model, lead to singular filtering models. More specifically, the magnitude of the observation noise may depend on the signal that makes the optimal filter singular. Some progress in this aspect was made recently.
It is the belief of this author that the time is ripe for a new textbook to reflect these recent developments. The main theme of this book is to recapitulate these advances in a succinct and efficient manner. The book can serve as a text for mathematics as well as engineering graduate and inspired undergraduate students. It can also serve as a reference for practitioners in various fields of applications. As noted, the aim of this book is to take the students to this exciting field of research through the shortest route possible. To achieve this goal, we completely avoid the chaos decomposition used in the classical filtering theory.
The main approach of this book is based on the particle representation for stochastic partial differential equations developed by Kurtz and Xiong. The methods used here can be applied to more general stochastic partial differential equations. Therefore, it provides a bridge to the readers who are interested in studying the general theory of stochastic partial differential equations.
We should mention that the propagation of chaos decomposition and the multiple stochastic integral methods have provided another type of numerical scheme in the approximation of the optimal filter. The advantage of this approach is that most of the computations are done “offline” in advance. This important development is not covered in this book because we want to limit the pre-requisite of this book. We only assume that the reader has basic knowledge of probability theory, for example, the material in the book of Billingsley.Brownian motion and martingales
Stochastic integrals and Itȏ’s formula
Stochastic differential equations
Filtering model and Kallianpur–Striebel formula
Uniqueness of the solution for Zakai’s equation
Uniqueness of the solution for the filtering equation
Numerical methods
Linear filtering
Stability of non-linear filtering
Singular filtering