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Stochastic Analysis with Lévy Noise in the Dual of a Nuclear Space.

Fonseca Mora, Christian Andrés (2015) Stochastic Analysis with Lévy Noise in the Dual of a Nuclear Space. PhD thesis, University of Sheffield.

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Abstract

In this thesis we introduce a new theory of stochastic analysis with respect to Lévy processes in the strong dual of a nuclear space. First we prove some extensions of the regularization theorem of Itô and Nawata to show conditions for the existence of continuous and càdlàg versions to cylindrical and stochastic processes in the dual of a nuclear space. Sufficient conditions for the existence of continuous and càdlàg versions taking values in a Hilbert space continuously included on the dual space are also provided. Then, we apply these results to prove the Lévy-Itô decomposition and the Lévy-Khintchine formula for Lévy processes taking values in the dual of a complete, barrelled, nuclear space. Later, we introduce a theory of stochastic integration for operator-valued processes taking values in the strong dual of a quasi-complete, bornological, nuclear space with respect to some classes of cylindrical martingale-valued measures. The stochastic integrals are constructed by means of an application of the regularization theorems. In particular, this theory allows us to introduce stochastic integrals with respect to Lévy processes via Lévy-Itô decomposition. Finally, we use our theory of stochastic integration to study stochastic evolution equations driven by cylindrical martingale-valued measure noise in the dual of a nuclear space. We provide conditions for the existence and uniqueness of weak and mild solutions. Also, we provide applications of our theory to the study of stochastic evolution equations driven by Lévy processes.

Item Type: Thesis (PhD)
Keywords: Cylindrical and Stochastic Processes, Dual of a Nuclear Space, Lévy Processes, Stochastic Integrals, Stochastic Partial Differential Equations
Academic Units: The University of Sheffield > Faculty of Science (Sheffield)
The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Identification Number/EthosID: uk.bl.ethos.675550
Depositing User: Mr Christian Andrés Fonseca Mora
Date Deposited: 15 Dec 2015 11:51
Last Modified: 03 Oct 2016 13:06
URI: http://etheses.whiterose.ac.uk/id/eprint/11362

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