The differential structure of spaces of rough paths

1 Brief description of the context of the research including potential impact
Rough paths are mathematical objects that have garnered significant attention in recent years thanks to the variety of applications they enjoy both in pure mathematics and applied fields. The space of rough paths is non-linear, which makes it difficult to understand its differential structure. While many theories about how to understand differentiability on non-linear spaces have been developed, a full account of what the differential structure of rough paths spaces has not been developed yet.

Smooth functions pervade mathematics, and the definitions of smoothness vary significantly. In real analysis, Ck spaces are spaces of smooth functions, where smoothness is defined as the continuity of the k-th derivative of the function. One important area where smooth functions are of crucial relevance is the study of differential equations. There, we can say that differential equations have solutions which belong to Ck for a certain k. Sobolev spaces are spaces of functions that, in a sense, generalise Ck spaces. While Sobolev spaces are spaces of smooth functions as much as Ck spaces, the requirement for a function to belong to a Sobolev space is that it has a certain degree of smoothness, but smoothness is understood in the weak sense. Sobolev spaces help us in the study of solutions to differential equations by allowing us to find weak solutions to certain partial differential equations in cases where there is no no strong solution, that is a solution that belongs to a Ck space.
Understanding the differential structure of rough paths will thus enable us to characterise Sobolev spaces on the space of rough paths. In turn, this will help us better understand the spaces in which to find solutions to certain classes of differential equations where rough paths play a crucial role: stochastic differential equations (SDEs).

2 Aims and objectives
The goal of this research project is to build an account of Sobolev spaces on the spaces of rough paths. Such an account has the potential to allow us to better characterise solutions to SDEs, which in turn have a broad range of pure mathematics and practical applications.

3 Novelty of the research methodology
The theories being developed are original since only a general account of the differential structure of spaces of rough paths has not been developed yet

This project falls within the EPSRC Mathematics research area, where Mathematical Analysis and Non-Linear Systems are some of the themes or 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