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

Self-similar Markov processes and applications

Group Probability and Stochastic Analysis
Supervisor
Description

Positive self-similar Markov processes (ssMps) have an important status as scaling limits of Markov chains, and since Lamperti's pioneering work have been strongly connected with Levy processes. They have enjoyed a resurgence in recent years on account of the newly discovered connection with Markov additive processes enjoyed by real and R^d-valued ssMps. There are a number of interesting open questions which are under investigation at the moment, in particular the study of conditionings of ssMps and the behaviour as the starting point approaches the origin. These form the focus of this PhD project, and they have further applications, for instance in the field of fragmentation.

 

References:

A. E. Kyprianou (2014) Fluctuations of Lévy processes with applications. Chapter 13. https://doi.org/10.1007/978-3-642-37632-0

 

Title

Growth-fragmentation phenomena

Group Probability and Stochastic Analysis
Supervisor
Description

Growth-fragmentation processes describe the evolution of a collection of cells which grow gradually and split apart (fragment) suddenly. They are motivated by applications to division of biological cells and of polymerisation phenomena, but they form a fascinating class of stochastic models in their own right. Pure fragmentation processes have been studied since Kolmogorov, and a framework for their understanding was introduced by Bertoin (2006). A much more recent development is the study of stochastic models incorporating growth. These were introduced in quite general form, again by Bertoin (2017), and there are many open questions, particularly concerning their behaviour at large times. A fascinating application of these processes is to models of 'random planar maps' which are motivated by statistical physics.

 

References:

J. Bertoin (2006) Random fragmentation and coagulation processes. https://doi.org/10.1017/CBO9780511617768

J. Bertoin (2017)Markovian growth-fragmentation processes. Bernoulli 23, no. 2, 1082–1101. https://doi.org/10.3150/15-BEJ770

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

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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