Optimal Stopping and Applications
This graduate course provides 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 roughly follows the treatment in Optimal Stopping and Free-Boundary Problems by Peskir and Shiryaev. A reasonable level of comfort with continuous-time processes and stochastic integration is assumed, although many techniques and ideas are introduced as they are encountered.
Lectured by Alex Cox and Andreas Kyprianou, with guest lectures from Juan Carlos Pardo Millan and Martin Herdegen. The course is timetabled at 13:15–15:05 on Tuesdays in 1W 3.20a.
Lecture notes
Syllabus
- 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 and 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 was expected that some volunteers would prepare some of the later lectures, and the list was updated as the semester progressed.