Ergodicity and Averaging of Fractional Dynamics

We are concerned with the long-time behavior of the solution to fractional-driven stochastic differential equations (SDEs). The interest in the field was sparked by a seminal article by Hairer where he built a sophisticated framework allowing to transfer most notions of classical Markov process theory. Given the existence and uniqueness of an invariant measure for fractional SDEs, the rate of convergence of the time-t law of the solution towards the invariant measure is of interest. The original work of Hairer obtained an algebraic rate for the additive noise case, which was later also established for multiplicative noise by Fontbona and Panloup for large and Deya, Panloup, and Tindel for small Hurst parameters, respectively. The fastest known convergence to the invariant measure (excluding the trivial case of an everywhere uniformly contractive drift) is due to recent work of Panloup and Richard for addtive noise. All of these rates are however significantly slower than for It\^o diffusions which often exhibit an exponential decay of appropriate metric. We thus see considerable potential for improvement of these results, both for additive and multiplicative noise.

Among many others, an important application exploiting ergodicity of stochastic processes is the study of averaging principles for fast-slow systems. With Birkhoff's theorem in mind, it is natural to expect that a system in ergodic fractional environment, which moves on a macroscopic scale, is well approximated by an effective, autonomous dynamics. To the best of our knowledge, there has been no previous study of such fractional multi-scale systems before and our results find applications in climate science, in which previous Markovian models produce predictions notoriously mismatching observational data.

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