On weak convergence of stochastic integrals and its applications

Many systems that are subject to uncertainty can be described using stochastic integration. Its relevance not only stems from its wide use as a theoretical tool within the field of stochastic analysis but also from a broad range of applications in areas such as physics, biology or mathematical finance. In many cases, there is a substantial need to characterize stochastic integral processes as a particular weak limit, either because this originates from the underlying model or due to the fact that certain desired properties cannot be established directly but will be transferred from the approximating sequence of processes.
Limit theorems in this context have been extensively studied throughout the second half of the 20th century - yet almost exclusively within the framework of Skorokhod's J1 topology. This topological structure on the space of càdlàg paths considers two functions to be close if they are close in the uniform norm after applying a marginal time shift. However, in recent years, the J1 topology has been seen to be too restrictive in a significant number of cases. Systems with more complicated interactions require us to work with weaker topologies. To name but a few, this includes continuous time random walks in statistical mechanics, microscopic models for supercooling in physics and contagion models for financial markets.
Indeed, consider for example a multitude of recurrent signals successively arriving and adding up into a large signal at an increasingly shorter delay before a specific time. Whereas convergence in this case fails under the J1 topology, Skorokhod's much lesser known M1 topology closes this gap by instead comparing distances of the connected graphs.

Taking into account this development, the research project in a first step aims at determining necessary and sufficient conditions for the weak convergence of stochastic integrals under Skorokhod's M1 topology and providing accessible criteria for applications. In particular, prevalent conditions for J1, which naturally involve the joint convergence in law of integrands and integrators as well as a firm uniform control on the variation of the integrators, will need to be extended by a certain regularity assumption additionally preventing the approximating integrands and integrators from presenting non-vanishing consecutive variation.
Among other applications, we expect such a result to be a useful tool for problems arising in the theory of functional limit theorems. In particular, it will contribute to identifying important central limit theorems for the aforementioned applications which are not in reach within the existing theory.
Further potential extensions may include an examination of the influence of already existing alternative topologies, such as the S topology or other Skorokhod topologies.

As a fundamental generalization to the described results for real-valued processes, we hope to establish a comprehensive mathematical framework allowing us to carry over these limit theorems to more general state space structures - and therefore opening up new fields of application such as for distribution-valued processes. The approach will additionally require finding novel techniques to deal with these generalized quantities.

This project falls within the EPSRC Mathematics Analysis, Statistics and Applied Probability, and Operational Research research areas.

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