Probability & Statistics
What we do
We have interests in both the development of new theoretical ideas and techniques within the realm of probability and stochastic models and the applications of probabilistic ideas to tackle novel problems within the real world.
Members of the group have long-term collaborations with leading research centres in the UK and abroad, as well as with colleagues at the University of Edinburgh and in other schools at Heriot-Watt University.
We work in a number of areas of applied and theoretical probability which are listed, in no particular order, below.
Limit theorems and approximations
In many models for real-world stochastic systems, probabilities and other characteristics of practical interest cannot be calculated exactly, perhaps because of a prohibitively complex model or large number of model components. One solution to this problem is to apply limit theorems, which give asymptotic estimates, or other approximations of quantities of interest. In a large number of settings we observe well-behaved estimates and approximations from one of a relatively small number of stochastic process, of which the Poisson and Gaussian processes are the best known examples. Central limit theorems, giving a Gaussian limit, are known to hold in a variety of settings, and a Poisson limit often occurs in the approximation of rare events. We work on limit theorems and approximations for a wide range of models and processes with a variety of dependence structures. Applications on which we have recently worked include communications and queueing systems, random networks, insurance, and Covid testing, among others. We have also worked on the theoretical underpinnings of several useful and widely applied approximation techniques.
Ergodicity of stochastic processes
We are interested in research problems related to the question of ergodicity of stochastic processes, with a focus on convergence of Markov processes to equilibrium. Examples of processes under consideration include solutions to stochastic differential equations, Markov chains and birth-death processes. We are interested in establishing conditions for existence of stationary distributions of such processes, as well as quantifying the convergence rate of the process to its equilibrium state. Besides their theoretical significance, such questions have multiple applications in fields such as computational statistics and machine learning, in problems of sampling (e.g. via Markov Chain Monte Carlo methods) and optimization (e.g. via stochastic gradient algorithms).
A random graph consists of a number of nodes, of which some randomly chosen pairs are connected by an edge. These are an essential tool in modelling communications, power and computer systems, social media contacts, the internet, and in many other areas. These
different applications give rise to a variety of random graph models with different mechanisms for deciding which nodes will be connected, and varying properties and characteristics. Some of these models are static, others can be thought of as evolving in time. Our work in this area includes the study of both exact and asymptotic properties of random graphs, including limit theorems and approximations. We have worked on understanding the structure and behaviour of different random graph models, including, for example, various measures of connectivity. These help us understand lengths of typical or extremal paths through the graph, or the likelihood of observing particular sub-graphs. We also study stochastic processes evolving on random graphs. Applications include the modelling of the spread of computer viruses through computer networks, spread of information through a social network or spread of infection through population.
Many interesting systems in physics and applied sciences consist of a large number of particles, or agents, (e.g. individuals, animals, cells, robots) that interact with each other. When the number of agents/particles in the system is very large the dynamics of the full particle system (PS) can be rather complex and expensive to simulate; moreover, one is quite often more interested in the collective behaviour of the system rather than in its detailed description. In this context, the established methodology in statistical mechanics and kinetic theory is to look for simplified models that retain relevant characteristics of the original PS by letting the number N of particles grow to infinity; typically, the resulting limiting equation for the density of particles is a low dimensional, (in contrast with the initial high dimensional PS), non-linear (stochastic) partial differential equation.
Beyond an intrinsic theoretical interest, such models were proposed with the intent to efficiently direct human traffic, to optimize evacuation times, to study rating systems, opinion formation, animal navigation strategies; in all these fields, as well as in stochastic simulation, they have been incredibly successful.
In this context, the research activity of our group revolves around the study of PSs modelled by stochastic dynamics (e.g. stochastic differential equations, continuous or discrete time jump processes etc ) whose limiting behaviour is described by either a deterministic PDE or by a stochastic PDE (SPDE) (for example equations of McKean-Vlasov type).
In addition to studying the expected behaviour of a system (say, law of large numbers or central limit theorem), one is often interested in the so-called rare events which have a vanishing probability but significant (sometimes disastrous) consequences (bankruptcy of a company, congestion in a network, blackout, etc.) It is of great importance to understand how likely such events are to occur, and also what is the most likely scenario for them to occur. We work on the theory of large deviations as well as its applications in energy, insurance, queueing networks.
We work on various stochastic networks arising from real-world applications including, but not limited to, biology, communication, data storage and processing, energy, healthcare, social networks. This is a multifaceted area of research where we study design, modelling, stability, control, performance, approximations and rare events in networks. The research involves development and use of techniques and tools from probability theory, stochastic processes, analysis, optimisation, PDEs, combinatorics and graph theory.
Stochastic modelling of biological systems
Work in this area includes: modelling endemic infections, in particular looking at the time until fade-out of infection, the endemic state prior to extinction, and the effects of different model assumptions upon these; the effects of population heterogeneities, eg pens within a pig farm, management groups within a dairy herd, or the existence of a group of "superspreaders"; applications of mathematical control theory in epidemic modelling.
The group has recently started research on the effects of migration on the spread of an epidemic. The research combines new and existing approaches and techniques from stochastic networks and contact processes.
Stochastic control and its applications
Our research focuses on developing novel theoretical and computational techniques to solve stochastic control problems arising from economics, finance and behavioural sciences. Our recent works include optimal control, time inconsistent stopping, time inconsistent LQ control and their applications in behavioural economics and insurance. We are particularly interested in deep learning algorithms for high dimensional time (in)consistent stochastic control problems.
Probability and computation
Stochastic models that are used in practice often need to be simulated computationally. Here, we have expertise in the simulation of stochastic differential equations and stochastic partial differential equations. We are particularly interested in efficient deterministic and random simulation strategies, such as the Multilevel Monte Carlo technique, low-rank methods,
sparse grid quadrature, and more general stochastic collocation methods. On the other hand, randomisation plays an important role in the modern numerical analysis of deterministic problems, e.g., optimisation, time stepping of differential equations, and approximation of matrices and matrix decompositions. Here, randomness is used for computational efficiency but also for robustness. We propose, analyse, and employ randomised techniques in numerical analysis. We study applications in finance, porous media, and data science.
Our work spans classical and Bayesian methods, and their application numerous areas. For example, the group has collaborated with scientists from a range of disciplines to develop Bayesian methods for problems arising in epidemiology, laser imaging, modelling of diabetes symptom reporting, and failure of drainage systems. The group also has a strong track record in classical techniques. Recent examples include the development of spline-based
methodology for smoothing mortality data and the use of classical filtering techniques to build fast emulators for complex building simulation models.
Panagiota Birmpa, Damian Clancy, Fraser Daly, Sergey Foss, Abdul-Lateef Haji-Ali, Jennie Hansen, Jonas Latz, Mateusz Majka, Simon Malham, Michela Ottobre, Seva Shneer, George Streftaris, Wei Wei, Anke Wiese