Stochastic differential equations with application to manifolds and nonlinear filtering

Abstract

This thesis follows a direction of research that deals with the theoretical foundations of stochastic differential equations on manifolds and a geometric analysis of the fundamental equations in nonlinear filtering theory. We examine the importance of modern differential geometry in developing an invariant theory of stochastic processes on manifolds, which allow us to extend current filtering techniques to an important class of manifolds. Furthermore, these tools provide us with greater insight to the infinite-dimensional nonlinear filtering problem. In particular, we apply our geometric analysis to the so called unnormalized conditional density approach expounded by M. Zakai. We exploit the geometric setting to study the geometric and algebraic properties of the Zakai equation, which is a linear stochastic partial differential equation. In particular, we investigate the use of Lie algebras and group invariance techniques for dimension analysis and for the reduction of the Zakai equation. Finally, we utilize simulation to demonstrate the superiority of the Zakai equation over the extended Kalman filter for a passive radar tracking problem.