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.

Title

Distributional approximation by Stein's method

Group Probability and Stochastic Analysis
Supervisor
Description

Stein's method is a powerful (and elegant) technique for deriving bounds on the distance between two probability distributions with respect to a probability metric.  Such bounds are of interest, for example, in statistical inference when samples sizes are small; indeed, obtaining bounds on the rate of convergence of the central limit theorem was one of the most important problems in probability theory in the first half of the 20th century.

The method is based on differential or difference equations that in a sense characterise the limit distribution and coupling techniques that allow one to derive approximations whilst retaining the probabilistic intuition.   There is an active area of research concerning the development of Stein's method as a probabilistic tool and its application in areas as diverse as random graph theory, statistical mechanics and queuing theory.

There is an excellent survey of Stein's method (see below) and, given a strong background in probability, the basic method can be learnt quite quickly, so it would be possible for the interested student to make progress on new problems relatively shortly into their PhD.  Possible directions for research (although not limited) include: extend Stein's method to new limit distributions; generalisations of the central limit theorem; investigate `faster than would be expected' convergence rates and establish necessary and sufficient conditions under which they occur; applications of Stein's method to problems from, for example, statistical inference.

Literature:

Ross, N. Fundamentals of Stein's method.  Probability Surveys 8 (2011), pp. 210-293.

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