Appears in collection : Introduction to Nonlinear Filtering Theory toward Particle Filtering

Nonlinear filtering is one of the main approaches to tackle many problems involving with fields such as financial mathematics, engineering sciences, fluid dynamics, physics, etc. Unfortunately, it is impossible to obtain analytical solutions for most of the stochastic nonlinear filtering models associate with the practical problems arising in above mentioned fields except small number of simple cases. Particle filtering plays an important role to solve such stochastic nonlinear filtering models numerically. In this short course, we are going to provide an introductory knowledge on nonlinear filtering theory and associated particle filtering approach. Firstly, we will define the nonlinear filtering problem and then step by step, we discuss the necessary tools (such as conditional expectation, Bayes formula, normalized and unnormalized conditional measures, Novikov’s condition, Girsanov transformation, Kallianpur-Striebel formula, etc.) to derive the measure-valued filtering equations known as Zakai equation and FKK equation. Then we discuss couple of particle filtering algorithms such as Monte-Carlo method, branching particle method, etc.