Filtering Equations

In this chapter, we shall introduce the most important results for continuous filtering problem. We will introduce stochastic partial differential equations satisfied by the filtered posterior distribution, which is well-known as the Kushner-Stratonovich (K-S) equation. We will use two different methods to derive this equation, namely, the change-measure method and innovation process method. In the change-measure method, we will introduce the conditional density formula for the continuous type, and we will derive an equation satisfied by a unnormalized conditional density function, which is well-known as the Duncan-Mortensen-Zakai equation. In application, observations are not a sequence of probability distributions but a specific sampled path. At the end of this chapter, we present the robust DMZ equation designed for path observation, which is used as a special change-measure method.