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. Examples of such
areas include audio engineering, finance and econometrics.
In general, the dynamical systems are not directly measureable and only incomplete
observations, quite often deteriorated by the presence noise are available. This leads us
to the main objective of stochastic filtering: the estimation of an evolving dynamical
system whose trajectory is modelled by a stochastic process called the signal, given the
information available through its partial observation.
Particle filters, which use clouds of weighted particles that evolve according to the law
of the signal process, can be used to approximate the solution of the filtering problem.
In time, as some of the particles become redundant, a procedure which eliminates these
particles and multiplies the ones that contribute most to the resulting approximation is
introduced at points in time called resampling/correction times. Practitioners normally
use certain overall characteristics of the approximating system of particles (such as the
effective sample size of the system) to determine when to correct the system.
There are currently no results to justify the convergence of particle filters with random
correction times to the solution of the filtering problem in continuous time. In this thesis,
we analyse particle filters in a continuous time framework where resampling takes place at
times that form a sequence of (predictable) stopping times. The particular focus will be
on the case where the signal is a diffusion process on a d-dimensional Euclidean space. We
will also look at central limit theorem type results for the approximating particle system.
The results will then be used to make inferences about the threshold used in the effective
sample size approach of approximating the signal.