Probability
Probability theory is broadly concerned with the development, analysis, and application of mathematical models describing randomness or 'noise'.
Research in probability theory in Bath concentrates on a number of models motivated by questions from physics, biology, finance and other applied fields. Current activities are organised under the umbrella of the Probability Laboratory at Bath (Prob-L@B), including lists of recent seminars and international visitors.
Our research in probability theory can be arranged under the following broad headings:
- Brownian motion and intersections of Brownian paths
- Branching processes and superprocesses
- Percolation and interacting particle systems
- Random graph models
- Lévy, positive self-similar Markov, and fragmentation processes
- Financial and insurance mathematics
- Self-organized criticality
Related areas of our research include:
Details of specific current research projects are given below.
Brownian motion and intersections of Brownian paths
Research staff: Alex Cox, Peter Moerters
Brownian motion is the unique stochastic process with continuous paths and independent, stationary increments. As such it is central to the study of probability theory and the basic building block for many other mathematical models in physics, finance and other applications.
Here is a simple problem related to one-dimensional Brownian motion. Given a probability distribution, can we find a stopping rule such that when Brownian motion is stopped using that rule, the value of the process on stopping is distributed like the given law. If we can find such a rule, can we find one to satisfy certain optimality properties? This is the so-called Skorokhod embedding problem, one of Alexander Cox's research interests. The answers to such questions are connected to a variety of other problems in both probability and finance.
One of the most fundamental questions about Brownian motion in dimensions two or more is whether two Brownian paths started in different points can have a non-trivial intersection, i.e. whether there are points in space, which are visited by both Brownian motions. Intersections of Brownian motions have now been studied for some time in probability theory and statistical mechanics. One of the reasons for this is that the properties of the intersections are analogous to those of many more complicated models in equilibrium statistical physics.
Fundamental results of Dvoretzky, Erdos and Kakutani in the 1950s show that such intersections exist with positive probability in dimensions two and three, but do not exist in higher dimensions. More recent results show that it is possible to define an intersection "local time" measuring the number of intersections between the paths. Determining the properties of this random variable is a challenging topic for research.
Branching processes and superprocesses
Research staff: Simon Harris, Andreas Kyprianou, Peter Moerters
Branching processes originated in the 19th century as models for family trees, and so it is not surprising that they are the building block for many probabilistic models for the growth of animal or plant populations. In many models of current interest the individuals may have different types, be in different spatial locations or interact in various ways with each or with an environment.
One of Simon Harris' and Andreas Kyprianou's research interests is in models that combine a branching model for population growth with a spatial model where each particle independently performs a Markov process (for example a Brownian motion, a general diffusion in n-dimensions or even a random walk). How quickly does the population grow? How fast does this cloud of particles colonise the surrounding empty space? Are there parts of space which particles of the process stop visiting? Does the count of particles in bounded domains grow asymptotically in a way that can be quantified deterministically? What happens if the breeding rate depends on position or type of the particle?
Peter Mörters is interested in spatial models of populations, which interact with a random environment. They study the behaviour of these populations in the long term, for example the question whether the population is clumping in certain regions of the space, and becomes extinct elsewhere.
There are strong links between the behaviour of spatial branching models and questions in the analysis of (non-linear) partial differential equations and special types of non-linear functional equations known as smoothing transforms. The models studied in Bath give valuable insight into the solutions of these equations.
Percolation and interacting particle systems
Research staff: Mathew Penrose
Typically, these systems involve a collection of particles living on a lattice, evolving in a random way, with neighbouring sites on the lattice interacting. Examples include certain models of spatial epidemics, and the sequential deposition of particles onto a surface. Such systems are naturally of consideration interest in physical chemistry and biochemistry. From a mathematical viewpoint there is substantial current interest in backing up the numerous existing simulation studies with some rigorous mathematical theory.
The related topic of percolation is a simple and popular stochastic model for disordered physical systems exhibiting phase transitions (i.e. sudden changes in the large-scale structure as a parameter is varied). Much of the work in Bath in this area is for continuum models, which have been growing in popularity, being often more realistic than lattice-based models. The contact process and oriented percolation are also interacting-particle systems that are considered to be branching random walk with interaction among individuals. This interaction is relatively week in high dimensions, which is related to a small intersection probability for two Brownian paths.
Random graph models
Research staff: Peter Moerters, Mathew Penrose
Consider a set of points placed at random in the plane, with a specified rule for connecting pairs of points which lie close to each other in order to construct a graph. For example, one could connect any two points separated by a distance of at most r. Or one could connect each point to its nearest neighbour, and there are many other ways to connect points. Such graphs arise in mathematical modelling, for example of communications or social networks, and statistical testing, for example in tests for uniformity of data. Mathew Penrose is involved in studying the properties of such graphs, in part motivated by issues of computational complexity and mathematical genetics.
Other random graph models of interest are defined dynamically: Starting from some initial configuration, at every time step we add a new vertex and connect it to the existing graph in a random fashion. In the preferential attachment models studied by Peter Mörters connections to old vertices are preferred if those have high degrees. This class of models is often used to model social interactions or the world-wide web.
Lévy, positive self-similar Markov, and fragmentation processes
Research staff: Andreas Kyprianou
Lévy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance is justified by their application in many areas of classical and modern applied probability. Examples of Lévy processes include Brownian motion and compound Poisson processes. However also included are much more exotic examples of stochastic processes. For example, processes that have paths which experience a countably infinite number of discontinuities over any finite time horizon. Further, processes that can be constructed in a such a way that, even in one dimension, with probability one, they never hit a prespecified point such as the origin. There are many other peculiarities of Lévy processes concerning their small and large time behaviour.
One of the main areas of research is understanding path properties of this large class of stochastic processes. Specific questions such as characterising the distribution of by how much a Lévy processes jumps over a prespecified level when it passes it for the first time. Also how can one can describe the dynamics of the current position of a Lévy process when measured from its last point of maximum? What is the long term asymptotic behaviour of a Lévy process when it is conditioned not to become negative in value?
These issues and many more concerning subtle aspects of the paths of Lévy processes turn out to provide a key to understanding quite intricate aspects of applications such as: the optimal time to sell a share whose dynamics are governed by an underlying Lévy processes, how to understand the behaviour of certain branching processes which can be expressed as time changed Lévy processes, understanding busy cycles and extremes of storage models whose input is described by the local time of a Lévy process spent at zero, modelling and controlling the wealth of an insurance company which experiences a high density of claims up to its moment of ruin, and so on.
Andreas Kyprianou is active in researching both analytic and probabilistic structures which appear as a consequence of path decompositions of Lévy processes and utilising them in classical and modern applied probability models such as those described above.
Positive self-similar Markov processes are a family of stochastic processes which are very closely related to Lévy processes. Fundamentally the two classes are related by a complex, but none the less explicit, invertible space-time transformation between their paths. As a result interesting features of, or new results for, Lévy processes have implications in the study of positive self-similar Markov processes and vice versa. 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 is interested in understanding how the multi-fragment random paths of such processes can be described via deterministic means. For example, can one look at fragments when they first become smaller than a certain constant size and then study the behaviour of this collection of fragments as this constant tends to zero. Thanks to many close connections between fragmentation processes and spatial branching processes a rich probabilistic variety of techniques are available for exploitation.
Financial and insurance mathematics
Research staff: Alex Cox, Andreas Kyprianou
A major aspect of modern financial markets are the trillions of dollars traded each year on a class of products known as derivatives. The trading of these finanical instruments hinges on connections to the study of stochastic processes made originally by Black and Scholes. The development of pricing techniques for these derivatives is intricately connected to the study of stochastic processes, and many fascinating connections are to be found between the two areas. The typical approach to the study of derivatives pricing is to assume that the financial assets behave according to a given model, however there is frequently uncertainty as to exactly which model is the right model to use; attempting to price and hedge risk under these conditions results in interesting classes of trading strategies, and Alexander Cox is interested in questions concerning these strategies.
Exotic options are a class of financial derivatives whose value depends on the path behaviour of the underlying risky asset. One popular class of model which is used for modelling risky assets is that of Lévy processes. Pricing of exotic options driven by this type of stochastic process quite often requires a quite deep understanding of mathematical phenomena coming from the theory of Lévy processes as well as other theories such as those of optimal stopping, stochastic games and optimal control. Andreas Kyprianou is interested in exploring the interplay between such phenomena and their relevance to the theory of pricing exotic options.
A classical field of study within an actuarial context concerns understanding how insurance companies become financially ruined as a consequence of sequence of claims which exceed the available wealth of the insuring agent. Andreas Kyprianou is interested in modern models of the wealth of insuring agents which are based on an underlying source of randomness taken to be (a functional of) a Lévy process. Ruinous scenarios can often be translated into barrier crossing problems for such processes which stimulates research into the mathematical subtleties of how and why ruin occurs.
Self-organized criticality
Research staff: Antal Jarai
There has been recent interest in probability in examples of stochastic processes where small disturbances of a system can cause huge effects leading to a dynamics involving abrupt changes rather than a smooth evolution. A common feature of such models - in the physics literature these occur under the term `self-organized criticality' - is that there is a local dynamics acting on a short time scale that leads to a very non-local dynamics on a longer time scale. The non-locality of the dynamics manifests itself in power law distribution of various characteristics of the system, akin to criticality in equilibrium statistical physics. It is a major challenge to rigorously prove the existence of such power laws. A fundamental example is the Abelian sandpile model, and others are the `forest-fire' model, invasion percolation, and the Bak-Sneppen model of punctuated equilibrium. Antal A. Járai's research in this area concerns the infinite volume limits of such models, primarily the Abelian sandpile.
