On the stochastic differential equations of filtering theory

摘要

We survey the development and role of stochastic differential equations in filtering theory. The basic stochastic differential equation of Fujisaki, Kallianpur, and Kunita, which solves the optimal filtering problem, is first derived, and then it is seen how the Zakai equation follows as a consequence. The Zakai equation is seen to yield a solution of the optimal filtering problem in the form of a conditional density, and it is shown how this equation, which is a stochastic partial differential equation, may be replaced by a nonstochastic partial differential equation which is continuously parametrized by the sample paths of the observation process; this latter equation yields a solution of the optimal filtering problem which is also robust with respect to modeling inaccuracies.