Particle Systems with Reflecting and Elastic Boundaries

Many systems that are subject to uncertainty can be described using particle systems. Examples reach over different subjects such as physics, neuroscience or mathematical finance. In this research project we consider a system of N particles described by stochastic processes on the positive half-line with a reflecting or elastic boundary at zero. Recent research has mostly focussed on the case of an absorbing boundary. A motivation for these models is to describe systemic risk in the banking systems or to price pools of many mortgages in a mortgage-backed security. Default appears if the associated particle hits the boundary and is absorbed. In some cases, mortgages are backed by the government and in case of a default the government guarantees the payments. Motivated by these cases we consider systems with particles that get reflected into the system if they hit the boundary. A relaxation of the reflecting case where defaults are always guarenteed the case of an elastic boundary can be considered. In this case default appears if payment does not happen for some time. As common for particle system the goal of the project is to analyse the population of particles by taking a limit as the number of particles grows to infinity. The finite particle system is described by an empirical measure process. The project aims to prove convergence in law of this process and to characterise the limit as a solution to a stochastic PDE. New boundary estimates are necessary to establish uniqueness results for this equation. This is caused by the difference in the nature of the boundary. An absorbing boundary offers additional speed of decay of the measure due to the vanishing density. When a reflecting or elastic boundary is considered, the density does not vanish when approaching zero. This requires finding novel ways of dealing with the measure decay at the boundary.
An important quantity in the case of reflecting boundary is the local time at the boundary. In the dynamic of the particles the local time appears as an additional term by using the Tanaka formula. It ensures that the particles stay in the domain. In the motivating example of government-backed mortgages the local time can be interpreted as a measure of how much money must be injected into the system to keep the mortgages from defaulting.
Further potential extensions include the consideration of a sticky boundary. This is a case that is between the reflecting and the absorbing boundary. A particle can stick to the boundary for some time before it gets reflected into the system. This will potentially create additional terms in the limit equation because the set of time spend at the boundary can have positive Lebesgue measure. It is motivated by the fact that mortgage owners need some time between the point at which the government steps in and the time they can start mortgage payments again.

Another interesting extension of these models arise when feedback is considered. In these extensions the drift of the particles is affected by the number of particles that have been absorbed at the boundary and pushes them closer to the boundary. The addition of feedback can lead to discontinuities in the loss process. This feature significantly complicates the mathematical analysis. Feedback models with an absorbing boundary have been used to model systemic risk in the financial system and a connection to the super-cooled Stefan's problem describing the evolution of the interface between liquid and solid face in super-cooled water has been shown. These models can be analysed be looking at McKean-Vlasov equation involving the law of the hitting time at the boundary. Interesting open questions include conditions that ensure that discontinuities appear or do not appear
and the analysis of feedback models with elastic boundaries.

EPSRC mathematical analysis research area

Probabilistic modelling permeates the Financial services, healthcare, technology and other Service industries crucial to the UK's continuing social and economic prosperity, which are major users of stochastic algorithms for data analysis, simulation, systems design and optimisation. There is a major and growing skills shortage of experts in this area, and the success of the UK in addressing this shortage in cross-disciplinary research and industry expertise in computing, analytics and finance will directly impact the international competitiveness of UK companies and the quality of services delivered by government institutions.
By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to
- sharpening the UK's research lead in this area and
- meeting the needs of industry across the technology, finance, government and healthcare sectors

MATHEMATICS, THEORETICAL PHYSICS and MATHEMATICAL BIOLOGY

The explosion of novel research areas in stochastic analysis requires the training of young researchers capable of facing the new scientific challenges and maintaining the UK's lead in this area. The partners are at the forefront of many recent developments and ideally positioned to successfully train the next generation of UK scientists for tackling these exciting challenges.
The theory of regularity structures, pioneered by Hairer (Imperial), has generated a ground-breaking approach to singular stochastic partial differential equations (SPDEs) and opened the way to solve longstanding problems in physics of random interface growth and quantum field theory, spearheaded by Hairer's group at Imperial. The theory of rough paths, initiated by TJ Lyons (Oxford), is undergoing a renewal spurred by applications in Data Science and systems control, led by the Oxford group in conjunction with Cass (Imperial). Pathwise methods and infinite dimensional methods in stochastic analysis with applications to robust modelling in finance and control have been developed by both groups.
Applications of probabilistic modelling in population genetics, mathematical ecology and precision healthcare, are active areas in which our groups have recognized expertise.

FINANCIAL SERVICES and GOVERNMENT

The large-scale computerisation of financial markets and retail finance and the advent of massive financial data sets are radically changing the landscape of financial services, requiring new profiles of experts with strong analytical and computing skills as well as familiarity with Big Data analysis and data-driven modelling, not matched by current MSc and PhD programs. Financial regulators (Bank of England, FCA, ECB) are investing in analytics and modelling to face this challenge. We will develop a novel training and research agenda adapted to these needs by leveraging the considerable expertise of our teams in quantitative modelling in finance and our extensive experience in partnerships with the financial institutions and regulators.

DATA SCIENCE:

Probabilistic algorithms, such as Stochastic gradient descent and Monte Carlo Tree Search, underlie the impressive achievements of Deep Learning methods. Stochastic control provides the theoretical framework for understanding and designing Reinforcement Learning algorithms. Deeper understanding of these algorithms can pave the way to designing improved algorithms with higher predictability and 'explainable' results, crucial for applications.
We will train experts who can blend a deeper understanding of algorithms with knowledge of the application at hand to go beyond pure data analysis and develop data-driven models and decision aid tools
There is a high demand for such expertise in technology, healthcare and finance sectors and great enthusiasm from our industry partners. Knowledge transfer will be enhanced through internships, co-funded studentships and paths to entrepreneurs