The ability to analyse, interpret and make inferences about evolving dynamical
systems is of great importance in different areas of the world we live in today.
Various examples include the control of engineering systems, data assimilation in
meteorology, volatility estimation in financial markets, computer vision and vehicle
tracking. In general, the dynamical systems are not directly observable, quite often
only partial information, which is deteriorated by the presence noise, is available.
This naturally leads us to the area of stochastic filtering, which is defined as the
estimation of dynamical systems whose trajectory is modelled by a stochastic process
called the signal, given the information accumulated from its partial observation.
A massive scientific and computational effort is dedicated to the development of
various tools for approximating the solution of the filtering problem. Classical PDE
methods can be successful, particularly if the state space has low dimensions (one to
three). In higher dimensions (up to ten), a class of numerical methods called particle
filters have proved the most successful methods to-date. These methods produce
approximations of the posterior distribution of the current state of the signal by
using the empirical distribution of a cloud of particles that explore the signal’s state
space.
In this thesis, we discuss a more general class of numerical methods which involve
generalised particles, that is, particles that evolve through spaces larger than the
signal’s state space. Such generalised particles include Gaussian mixtures, wavelets,
orthonormal polynomials, and finite elements in addition to the classical particle
methods. This thesis contains a rigorous analysis of the approximation of the solution
of the filtering problem using Gaussian mixtures. In particular we deduce
the L2-convergence rate and obtain the central limit theorem for the approximating
system. Finally, the filtering model associated to the Navier-Stokes equation will be
discussed as an example
