About the Course in Spring 2011 --- It's All New!
This course covers different topics each year. If this year's topics are of interest to you and you have already taken Stat 900, you can sign up for this year's course under an independent study number.
This is an advanced course but it is expected that students will come from diverse backgrounds and have a wide range of interests, so the actual technical requirements are kept pretty light. This is a course that is designed to be inclusive. People with very different skill sets should still find plenty of material that is both interesting and accessible. You will see that the course deliverables are designed to accommodate this diversity.
What's Covered This Year?
This year we will deal with topics in both discrete time and continuous time. The over arching theme will be the relationship between probability and optimization. We will be interested both in algorithms and in concrete mathematical solutions (i.e. exact formulas --- when we can get them!).
In particular we will cover:
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Problems of Optimal Stopping and Optimal Sequential Selection. The most classic of these is the famous "Secretary Problem" but this is just the tip of the ice burg. We'll cover some of the most interesting bits of the famous book The Theory of Optimal Stopping by Chow, Robins and Siegmund.
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Stochastic Control Models with Fixed Costs, or what Nancy Stokey calls "The Economics of Inaction." This is really a large class of economic models, most of which are simple from the modeling point of view, yet still interesting from the mathematical point of view. We will deal with many parts of Stokey's book (which is recommended but not required).
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Multi-Armed Bandit Problems. In the classic case one has to choose between two "slot machines" (devices with fixed payout distributions that are unknown to the player). The task is to go back and fourth between the two machines in a way that maximizes your long term total return from playing. "Bandit problems" can be used to model many other kinds of problems.
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Tools to be Developed --- A Subsample!
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Wald Lemma and related tools of Martingale Theory (elemenatary but very useful --- also subtle when done professionally)
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Ito's Formula and the working tools of stochastic calculus (This would be a course except that we favor cool calculations over theory building)
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The Gittin's index (useful but often isolated, maybe we can bring it into a richer context)
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Prophet Inequalities --- Surprisingly often you can do almost as well without knowledge of the future as you can do with knowledge of the future. We'll look at some original (but relatively easy) papers.
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Techniques of mathematical problem solving --- especially as they relate to probability and optimization.
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Processes and Procedures:
There will not be regular homework or tests, but instead students are expected to develop a project. This need not be a genuine "research project" but it should reflect the skills that one needs to get up-close and personal with an honest research problem.
The main deliverable will be a final project paper of approximately twenty pages. There will be preliminary deliverables of a project proposal, a project progress report, and a one-half hour class room presentation on the material of the project.