Lee, Rui Xin (2013) The existence and characterisation of duality of Markov processes in the Euclidean space. PhD thesis, University of Warwick.

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Abstract

This thesis examines the existence of dualMarkov processes and presents the full characterization of Markov processes in Euclidean space equipped with the natural order (the Pareto order).

Considering the theory of Siegmund’s duality for real-valued Markov Processes,we have presented an alternative proof to Siegmund [72] using Lebesgue–Stieltjes integration by parts to show the existence of a Markov dual process in several one dimensional cases, including the real space and closed intervals. Assuming that a dual process exists, we also provided a straightforward method, using duality relation, to compute explicitly the dual generator to a Feller process of the usual Lévy-Khintchine type.

We extended Siegmund’s duality to finite dimensional space equipped with the Pareto order. The existence of a dual Markov process on an arbitrary Euclidean space is shown using Fubini's theorem applied to Siegmund's approach. Given a pre-generator of the general Lévy-Khintchine type, we were able to construct a Feller process with an invariant core under some conditions assumed on the pre-generator. Furthermore, we also showed the criterion for the Feller process to have a dual Markov process.

We then studied the relationship between intertwining and duality for two processes in the sense of Ef(Xxt,y) = Ef(x,Yyt) for a certain function f. Of most interest are shift-invariant functions (functions which depend on the difference of their arguments). To explore this, we developed a systematic approach to duality using the analysis of the generators of dual Markov processes, then illustrated this approach using various examples. In particular, we gave a full characterization of duality arising from Pareto order in Rd in terms of generators for basic classes of Feller processes. Lastly, we initiate the application of intertwining to the study of duality of Markov processes in domains with a boundary. To circumvent specific difficulties arising from the boundary, we introduce an additional tool of a regularized dual.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Markov processes
Official Date:	September 2013
Dates:	Date Event September 2013 Submitted
Institution:	University of Warwick
Theses Department:	Department of Statistics
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Kolokoltsov, V. N. (Vasiliĭ Nikitich)
Extent:	viii, 100 leaves
Language:	eng

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