Department of Mathematical Sciences


(Probability Laboratory at Bath)

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Ph.D. Students

  • Chris Daniels
  • Samuel Gamlin
  • Horacio Gonzalez
  • Maren Eckhoff
  • Francis Lane
  • Marion Hesse
  • Christoph Höggerl
  • Steven Pagett
  • Istvan Redl
  • Benjamin Willey

Previous members

List of possible Ph.D. project areas

  • Branching diffusions and reaction diffusion equations

    A variety of branching diffusion models and their links with certain non-linear partial differential equations will be investigated. The project will involve many fundamental areas of modern probability theory, including Brownian motions, Martingales and Markov processes.

  • Skorokhod embedding problem

    The Skorokhod Embedding problem can be summarised as follows: given a measure and a Brownian motion, can we find a stopping time such that the stopped Brownian motion has the given distribution. The problem was initially solved by Skorokhod in the 1960s, and there has since been a number of different solutions in the literature. There has also been considerable recent interest, motivated both by connections to Mathematical Finance, and to Optimal Stopping. A project in this area, supervised by Alex Cox, could consider both theoretical and applied aspects of the topic.

  • Robust techniques for pricing and hedging options

    The classical approach to the pricing and hedging of financial options - pioneered by Black and Scholes, is to postulate a model for the underlying asset, and use the notion of risk-neutral pricing to find the arbitrage-free price of derivative contracts based on the underlying. The recent financial crisis has demonstrated the frailty of such model based techniques, and highlighted the need for more robust methods of hedging and pricing derivative contracts. A starting point for an alternative approach is to ask: given a set quoted prices, when are these prices consistent with some model? This proves to be the starting point for a number of interesting questions, and a project in this area, supervised by Alex Cox, would look to investigate some of these.

  • Random sequential particle deposition

    Suppose particles are deposited sequentially at random onto a surface. If the particles are flat, the ultimate density at which they fill the surface is of interest. If they are three-dimensional, they stack up on each other and the shape of the growing interface is of interest. For scientific background, see the 1993 survey article by J. Evans, Reviews in Modern Physics 65, 1281-1329; for recent rigorous results, see Penrose and Yukich, Annals of Applied Probability 12, 272-301. It would be of interest to try to add to these. Random geometric graphs and communications networks Random geometric graphs are constructed by placing points at random in a planar region and connecting nearby points. They are of considerable interest in telecommunications. My recent book "Random Geometric Graphs" (OUP) sets out a mathematical theory of these graphs, but there remain many open questions.

  • Abelian sandpiles

    The Abelian sandpile (also known as the chip-firing game) is a Markov chain describing the evolution of particle configurations on a finite graph. Close variants of the model were discovered independently in theoretical physics and combinatorics. A lot of interesting mathematics comes together in its analysis: Abelian groups (the "Sandpile Group"), spanning trees, random walk, loop-erased random walk, and determinants. For a brief introduction and references, see Antal Jarai's homepage. A number of PhD topics are available in this area.

  • Fragmentation and coalescent processes

    Fragmentation processes are stochastic processes that describe how an arbitrary object falls apart over time by a sequence of random dislocations (for example as one sees in rock crushing). In their most general form, fragmentation processes allow for an countably infinite number of fragments at any moment of time as well as countably infinite number of dislocations over any finite time horizon. Built into the structure of such models are features of infinite divisibility and self-similarity which are very closely related to the theory of Lévy processes and positive self-similar Markov processes as well as certain exchangeability phenomena that appear in the combinatorial theory of random partitions. One may also consider random processes which perform the opposite action to a fragmentation process, namely they coalesce mass. Andreas Kyprianou and Simon Harris are interested in how to characterize the evolution of fragments as the dislocate or coalesce through time as well as their connection to certain deterministic integral equations.

  • Lévy processes, positive self-similar Markov processes and applied probability

    A variety of projects are possible which, at their core, concern subtle path and stochastic analytic properties of Lévy processes. Closely related are positive self-similar Markov processes whose paths may be seen via a space-time transformation of the paths of Lévy processes. Recent projects of A. Kyprianou include topics in which the latter two families of stochastic processes play a predominant role such as: financial mathematics (exotic option pricing and credit risk), optimal stopping problems, stochastic games, stochastic control problems, risk insurance models, heavy-tailed phenomena, fluctuations of stable processes, Markov additive processes, local time and storage models, asymptotic behaviour of conditioned Lévy processes and random walks as well as the behaviour of population models known as continuous state branching processes. To get an overview of the theory and application of Lévy processes, one may consult Prof. Kyprianou's recent book: "Introductory Lectures on fluctuations of Lévy process with applications" Universitext, Springer, 2006

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