## An Introduction to Stochastic Filtering TheoryStochastic Filtering Theory uses probability tools to estimate unobservable stochastic processes that arise in many applied fields including communication, target-tracking, and mathematical finance.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 although still in its infancy, the study of singular filteringmodels has yielded exciting results.In this text, Jie Xiong introduces the reader to the basics of Stochastic Filtering Theory before covering these key recent advances. The text is written in a style suitable for graduates in mathematics and engineering with a background in basic probability. |

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### Contents

1 Introduction | 1 |

2 Brownian motion and martingales | 15 |

3 Stochastic integrals and Itôs formula | 36 |

4 Stochastic differential equations | 61 |

5 Filtering model and KallianpurStriebel formula | 82 |

6 Uniqueness of the solution for Zakais equation | 96 |

7 Uniqueness of the solution for the filtering equation | 121 |

8 Numerical methods | 132 |

9 Linear filtering | 157 |

10 Stability of nonlinear filtering | 186 |

11 Singular filtering | 231 |

255 | |

List of Notations | 266 |

269 | |

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Applying Itô's formula approximation asymptotically stable bounded Brownian motion Cauchy–Schwarz inequality chapter constant K1 Corollary d-dimensional Brownian motion define Definition denote easy to show equation 4.1 exists a constant filtering equation filtering model filtering problem follows from equation formula to equation function hence independent Kalman–Bucy filter Lemma linear Lipschitz continuous local martingale mapping Markov process Markov property Meyer’s process MF(S o-field observation process optimal filter particle system probability measure probability space Proof Let random variable respect Riccati equation right-continuous satisfies the following sequence solution to equation SPDE square-integrable square-integrable martingale stability ſto stochastic basis stochastic differential equation stochastic integral stochastic process submartingale subspace Suppose system equation unnormalized filter vectors Wasserstein metric Zakai’s equation