This thesis addresses some aspects of the connection between convex analysis andoptimal stopping and control problems. The first chapter contains a summary of theoriginal contributions made in subsequent chapters. The second chapter uses elementary tools from convex analysis to establish anextension of the Legendre transformation. These results complement the results in[66] and are used to provide an alternative proof that Nash equilibria exist in optimalstopping games driven by diffusions. In the third chapter a ‘maximum principle’ for singular stochastic control is es-tablished using methods from convex analysis which is a generalisation of the firstorder conditions derived in [18]. This ‘maximum principle’ is used to show that thesolution to certain singular stochastic control problems can be expressed in termsof a family of associated optimal stopping problems. These results connect the firstorder conditions in [3] and the representation result originating in [5] to variationalanalysis. In particular, the Legendre transform is used to derive first order conditionsfor a class of constrained optimisation problems. Sections 2.1-2.4 and Example 30 have been accepted for publication to the ‘Journalof Convex Analysis’ as [75] subject to minor corrections. The suggested revision hasbeen implemented in this thesis.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:603151 |
| Date | January 2014 |
| Creators | Sexton, Jennifer |
| Contributors | Peskir, Goran |
| Publisher | University of Manchester |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://www.research.manchester.ac.uk/portal/en/theses/optimal-stopping-and-control-problems-using-the-legendre-transform(aa3ce911-2a1d-4d48-8096-367706c798c9).html |
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