Course No.：21615Z
Period：10
Credits：0.5
Course Category：Lecture
Primary Coverage：
Talks 1 -2 Stochastic Filtering Model and Kallianpur-Striebel formula
We first introduce a few motivating examples arising from various fields of applications: signal estimation for wireless communication, portfolio optimization in mathematical finance, and pollution detection for environment protection. Then, we will introduce the filtering model to be studied. The Kallianpur-Striebel formula will be established which will be used to derive the stochastic partial differential equations (SPDEs) satisfied by the optimal filter and its unnormalized version.
Talks 3-4 Existence and Uniqueness for the solutions of SPDEs
First, we shall study the existence and uniqueness for the solution to a linear SPDE satisfied by the unnormalized filter by some Sobolev space type estimates. Then, we consider the same problem for the nonlinear SPDE satisfied by the optimal filter. To this end, we shall study an interacting particles system governed by an infinite system of stochastic differential equations (SDEs). Finally, we derive the existence and uniqueness for the solution of the nonlinear SPDE from that of the infinite system of SDEs and that of a corresponding linear SPDE. Although we study such problems under the filtering setting, the techniques introduced can be applied to more general nonlinear SPDEs.
Talk 5 Numerical approximations for the optimal filter
The explicit formulas to the solution of the filtering problem are rarely available. In this talk, we introduce a numerical scheme based on a branching particle system. We will prove the convergence of such scheme and characterize the rate of convergence.
References:
1. Xiong, J. (2008). An Introduction to Stochastic Filtering Theory, Oxford Graduate Texts in Mathematics. 18. Oxford University Press.
2. G. Kallianpur (1980). Stochastic Filtering Theory. Springer-Verlag.
Author：Jie Xiong 
