Probability and Stochastic Analysis PhD projects

This page provides a (partial) list of specific (and not so specific) PhD projects currently offered around the Probability and Stochastic Analysis topic.

Identifying an interesting, worthwhile and do-able PhD project is not a trivial task since it depends crucially on the interests and abilities of the student (and the supervisor!) The list below therefore contains a mixture of very specific projects and fairly general descriptions of research interests across the School. In either case you should feel free to contact the potential supervisors to find out more if you're interested.

You should also explore the School's research groups' pages, and have a look around the homepages of individual members of staff to find out more about their research interests. You may also contact us for general enquiries.

Title

Scheduling and Parallel Computing

Group Probability and Stochastic Analysis
Supervisor
Description

Consider a large computational task, for instance, solving a large system of linear equations. This task can be split into many smaller jobs which are then scheduled and queued at a large number of different heterogeneous computing resources and are executed in parallel. The processing requirements of the different resources (for instance CPUs and GPUs) are different, and they may well have different communication costs. In this project we will be interested in understanding the stochastic effects before designing (and implementing) novel distributed scheduling algorithms.

We seek a student with skills in probability, optimization and mathematical modeling. A student with good programming skills (particularly in C/C++) would be preferred.

Title

Optimal prediction problems driven by Lévy processes

Group Probability and Stochastic Analysis
Supervisor
Description

An optimal prediction problem is a type of optimal control problem very similar to classic optimal stopping problems, however with the crucial difference that the gains process is not adapted to the filtration generated by the driving process. Consider for instance the problem of stopping a Brownian motion at a time closest (in some suitable metric) to the time the Brownian motion attains its ultimate supremum. This is an optimal prediction problem: the gains are determined by the difference between the chosen stopping time and the time at which the Brownian motion actually does attain its ultimate supremum, and as the latter quantity is not known until the whole path of the process is revealed the gains is indeed not adapted.

Besides being mathematically very appealing, optimal prediction problems have many applications, for instance in mathematical finance, insurance, medicine and engineering, to name a few. They were relatively recently introduced, and the bulk of the work done so far is for diffusions only. In this project, several types of optimal prediction problems driven by Lévy processes rather than diffusions will be studied. Lévy processes play a role in mathematical finance as an extension of the classic Black & Scholes model and in insurance as an extension of the classic Cramer-Lundberg model for instance.

Note that this project requires a very strong background in probability theory, in particular stochastic processes, and mathematical analysis. Preferably the student should already be familiar with Lévy processes. Experience with computer programming and implementing numerical schemes is very helpful.

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