Blackwood, Stefan (2014) Lévy Processes and Filtering Theory. PhD thesis, University of Sheffield.

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Abstract
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 nonlinear 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 alphastable 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 nonlinear 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 KushnerStratonovich equation. Finally we prove the uniqueness of solution to the Zakai equation, which in turn leads to the uniqueness of solution to the KusherStratonovich equation.
Item Type:  Thesis (PhD) 

Keywords:  Lévy process, Kalman filter, Zakai equation, Stochastic filtering theory 
Academic Units:  The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) 
Identification Number/EthosID:  uk.bl.ethos.605526 
Depositing User:  Stefan Blackwood 
Date Deposited:  04 Jul 2014 08:43 
Last Modified:  03 Oct 2016 11:16 
URI:  http://etheses.whiterose.ac.uk/id/eprint/6380 