An exploration of quasimartingales, their applications and related topics

Stochastic processes are one of the central objects in the field of stochastic analysis and are used in numerous applications. Among these so-called stochastic processes is a kind known as quasimartingales. First introduced in 1965, any quasimartingale can be expressed as the difference of two unique stochastic processes which are 'more simple' to understand, known as non-negative supermartingales. This difference is known as the Rao decomposition of a quasimartingale. By studying quasimartingales, one could not only broaden the knowledge of a substantial though not very widely studied area in stochastic analysis, but further enhance studies in other fields through potential applications of quasimartingales.

Broadly speaking, the aim of this project is to further research and study quasimartingales. More specifically, there are two initial objectives. The first would be to find a new, fast way to determine the Rao decomposition of a general quasimartingale. The second would be to study the space quasimartingales when viewed as an object called a Riesz space. There is a vast catalogue of interesting properties that so-called Riesz spaces may possess, and one would seek to determine which of these properties the space of quasimartingales possesses, and which it does not. Later on, one would seek to find various applications of quasimartingales, whether these be ones new to existing literature (such as the growing fields of rough path theory and optimal transport) or extensions of already known applications. Such applications include finance, control theory, and studies in stochastic integration and statistical causality.

To determine a way to find the Rao decomposition of a quasimartingale, we will determine an algorithm that would return this decomposition. To the best of our knowledge, an algorithm to determine the Rao decomposition of a general quasimartingale does not exist. In fact, there is only one known procedure for finding the said decomposition (for an arbitrary quasimartingale). Furthermore, we will study an algorithm first published by Guido Ascoli that has almost never been studied since its publication in the 1930s. This algorithm expresses certain (more precisely, finite variational) functions as the difference of two non-decreasing functions, and we devise a 'stochastic' analogue of this algorithm.
Also, we seek to use the framework of Riesz spaces to determine the correctness of the decomposition yielded by the algorithm to be proposed.
In some key steps of the proof of the validity of our proposed algorithm, we will use techniques usually used in the field of optimal stopping that have never been used when studying quasimartingales.

This project falls within (but is not necessarily limited to) the EPRSC research areas of 'mathematical analysis' and 'statistics and applied probability'. This project falls more precisely within the area of stochastic analysis. The study of a relatively little known subject (compared to subjects therein) in stochastic analysis would help to broaden even further what is already a thriving field. Through the project's close connection with stochastic analysis, the project could contribute some more tools in the study of statistics and applied probability. This contribution would deepen if we are able to expand the link between quasimartingales and statistical causality, or if we even find new applications of quasimartingales to specific subjects in this research area.
Through the connection of the project with the two research areas, the project would contribute to the mathematical powerhouse that EPSRC continues to build.
As the work in the project makes use of relatively unknown literature, the project would additionally help to minimise 'waste' of worthwhile yet seldom studied literature, thereby contributing to the establishment of an effective ecosystem of research, which is a priority of EPSRC.

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