Conditioned random dynamics

Absorbing Markov processes is a branch of the general theory of Markov processes theory. The theory of absorbing Markov processes aims to study a Markov process X_t with state-space E = M U d, such that d is an absorbing state for X_t. This means that if X_t lies in d, then X_(t+s) lies in d for all s>0. Such a theory aims to understand the behaviour of the paths that do not escape from M for a long time.

Absorbing Markov chains appears nConditioned random dynamicsaturally in several events in nature. For instance, in biology, an absorbing state can be viewed as the extinction of species or the disappearing of characteristics in a population. Most games of chance can be modelled as absorbing Markov processes, where the absorbing state is defined as the cash reserve of a player being equal to zero. In medicine, an absorbing state can be defined as a condition reached by the body where it cannot cure or regenerate anymore.

The theory of absorbing Markov processes for continuum state spaces are still in early stages of development, and some fundamental open problems remain. One of the main techniques of this area is based on the existence of quasi-stationary and quasi-ergodic measures on M. The concept of a quasi-stationary measure is the analogue of stationary measures to Markov processes with escape. On the other hand, quasi-ergodic measures are a natural extension of the concepts of ergodic measures and Birkhoff sums to the context of absorbing Markov processes

In the current stage of the theory, there exist several elementary absorbing Markov processes for which the existence of quasi-stationary and quasi-ergodic measures cannot be guaranteed. With this technical difficulty in mind, a central question in absorbing Markov processes theory concerns the existence and uniqueness of quasi-stationary and quasi-ergodic measures. Despite increasing attention, sufficient conditions have {remained} quite restrictive in a general context.

One of the main aims of this project is to create tools that can be used to qualitatively study an absorbing Markov process. In the current stage of the research, we were able to prove the existence and the uniqueness of quasi-stationary and quasi-ergodic measures for a large class of absorbing Markov processes under relatively weak hypotheses, thus substantially extending the conditions in which quasi-stationary and quasi-ergodic measures are known to exist.

These results were developed analysing the transition probability function of an absorbing Markov process as a bounded linear operator, which is well-behaved from a Banach lattice point of view. This allows us to describe its spectrum and consequently construct the desired measures. To the best of our knowledge, we are the first to employ results from Banach lattice theory in this context, which may also turn out to be a powerful tool for future developments.

This project has collaborators, Dr Jeroen Lamb, Dr Martin Rasmussen and Dr Guillermo Olicón Méndez.

This project falls within the EPSRC statistics and applied probability 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