PSD - Prof John Moriarty

Optimal stopping in higher dimensions

Supervisor: Prof John Moriarty

Project description:

“Optimal Stopping” (OS) is a mathematical theory that can help you decide when to stop looking and make a choice, such as when to buy a house. The theory is based on the idea that extra information is unlikely to be useful after a certain point. In the context of buying a house, for example, the theory suggests that you should inspect 37% (or 1/e) of the available properties and then choose the next one that is better than these.

More mathematically, a striking connection between OS and convex geometry was made in the 1960s by Dynkin and Yushkievich, who solved one-dimensional optimal stopping problems. However no such result exists for two or higher dimensional OS problems, and this project aims to begin addressing this gap.

Concretely, the objectives of the project are:

1. Extension of geometric solutions to two-dimensional problems: Which class of two-dimensional OS problems can be solved by extending the one-dimensional geometric approach?
2. Algorithmic construction of the solutions in 1.: Can the solutions identified in 1. be obtained by an efficient numerical algorithm? Alternatively, can useful bounds on the solutions be obtained efficiently?
3. Extension of 1. and 2. to risk-sensitive problems: Sensitivity to risk is important in many contexts. Can the solutions in 1. and 2. also be obtained when the problem is made risk-sensitive? 

The successful candidate will work on the above objectives, and there will also be opportunities to work on new (but related) directions, such as broader classes of stochastic optimisation problem related to OS. The candidate is expected to have a strong background in continuous-time stochastic processes and optimisation. Prior knowledge in partial differential equations / potential theory, and experience with algorithm design and implementation, would be advantageous.

Further information: 
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