Trawl processes

This research project lies at the intersection between pure mathematics and Spatio-temporal statistics and provides stochastic models which in turn can offer insights into physical phenomena such as turbulence, financial markets, and much more. The overall goal is to model complex dynamics in a way that incorporates available knowledge about the system of interest.

In summary, this project concerns trawl processes: continuous-time, stationary and infinitely divisible processes that can describe a wide range of possible serial correlation patterns in data. Trawl processes are studied by the general topic of Ambit Stochastics, which has first been developed to model physical phenomena such as turbulent flow and tumor growth. Nevertheless, since its introduction, Ambit Stochastics proved to be a powerful modeling tool in many other settings, such as Spatio-temporal statistics, brain imaging, and finance.


The first step is to develop the class of trawl processes in time, space, and space-time.
We want to characterize their path properties, derive stochastic simulation algorithms, assess
the numerical accuracy of these simulations algorithms by deriving theoretical error bounds and implement the simulation algorithms in R/Python. Deriving simulation schemes and understanding their theoretical and numerical errors comes naturally as a first step, before employing such processes to model real-world data.


Next, we would like to explore parameter estimation for such processes. Previous research considered the generalized method of moments and, for the subset of integer-valued trawl processes, the maximum composite likelihood method. We would like to investigate the performance of these methods for general trawl processes. Once suitable methods have been developed to simulate the trawl process and infer their parameters, we can proceed with simulation-based inference methods and perform probabilistic forecasting at any time horizon.

Finally, while trawl processes can be used directly to model dynamical systems of interest, they can also be incorporated in a hierarchical model, as a stochastic volatility component or otherwise. A natural candidate in that sense is the class of Exponentiated Trawl Processes(ETP). We would then like to compare the performance of these models with that of OU, supOU processes, which do not share the same scaling properties as ETP. We will then study how the ETP parameters can be estimated when ETPs are not directly observed but are latent stochastic volatility processes, as described above.
This project falls within the EPSRC Applied Probability and Statistics research area and has been supported by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation (EP/S023925/1). There are no companies involved in this research project as of September 2021. There is a collaboration with the Technical University of Munich.

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