Optimal Stopping and Applications

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This graduate course aims to provide an introduction to the theory and practice of optimal stopping. Optimal stopping questions arise in a number of practical and theoretical situations, wherever there is the need to make a single, irreversible decision under uncertain knowledge of the future. Applications of the theory arise in statistics, mathematical finance and stochastic analysis. The study of optimal stopping problems leads to connections with free-boundary problems.

The course will roughly follow the treatment given in the book Optimal Stopping and Free-Boundary Problems by Peskir and Shiryaev. There is a copy of this book in the library (currently on order), and from campus one can access the book electronically. A reasonable level of comfort with continuous-time processes and stochastic integration will be assumed, although many techniques and ideas will be introduced as encountered.

UPDATED: The course is timetabled at **13.15-15.05** on Tuesdays in 1W3.20a. An approximate outline of the lectures is as follows:

Lecture 1 (27/1/09)
Introduction; Motivation; Optimal Stopping problems in discrete time
Lecture 2 (3/2/09)
Optimal Stopping problems in discrete time; Martingale approach; Markovian Approach
Lecture 3 (10/2/09)
Optimal Stopping problems in continuous time; Martingale approach
Lecture 4 (17/2/09)
Optimal Stopping problems in continuous time - Markovian Approach
Lecture 5 (24/2/09)
Free-boundary problems and optimal stopping
Lecture 6 (3/3/09)
(Andreas Kyprianou) Smooth fit & continuous fit
Lecture 7 (10/3/09)
(Andreas Kyprianou) Example: Lévy processes
Lecture 8 (17/3/09)
(Juan Carlos Pardo Millan) The Gapeev-Kühn stochastic game driven by spectrally positive Lévy processes.
Lecture 9 (24/3/09)
(Martin Herdegen) Example: Mathematical Finance - The American Put
Lecture 10 (21/4/09)
Example: Optimal stopping of the maximum

It is expected/hoped that some volunteers will prepare some of the later lectures, and the list will be updated as the semester progresses.

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Last updated: 12/1/09
Maintainer: Alex Cox