Lévy Processes and Filtering Theory

Stochastic filtering theory is the estimation of a continuous random system given a sequence of partial noisy observations, and is of use in many different financial and scientific areas. The main aim of this thesis is to explore the use of Lévy processes in both linear and non-linear stochastic filtering theory. In the existing literature, for the linear case the use of square integrable Lévy processes as driving noise is well known. We extend this by dropping the assumption of square integrability for the Lévy process driving the stochastic differential equation of the observations. We then explore a numerical example of infinite variance alpha-stable observations of a mean reverting Brownian motion with a Gaussian starting value, by comparing our derived filter with that of two others. The rest of the thesis is dedicated to the non-linear case. The scenario we look at is a system driven by a Brownian motion and observations driven by an independent Brownian motion and a generalised jump processes. The result of our efforts is the famous Zakai equation which we solve using the change of measure approach. We also include conditions under which the change of measure is a martingale. Next, via computing a normalising constant, we derive the Kushner-Stratonovich equation. Finally we prove the uniqueness of solution to the Zakai equation, which in turn leads to the uniqueness of solution to the Kusher-Stratonovich equation.