Optimal Stopping and Control Problems using the Legendre Transform

  • Jennifer Sexton

Student thesis: Phd

Abstract

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.
Date of Award1 Aug 2014
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorGoran Peskir (Supervisor)

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