Wasserstein distributional sensitivity to model uncertainty in dynamic context

Stochastic optimization problems in a multi-period (dynamic) setting are a staple of applied mathematics in many domains. In particular, they are the backbone of quantitative finance and financial economics, allowing us to tackle the problems from optimal investment decisions, through hedging problems to equilibrium pricing, and more. In such context, the model is usually derived from theoretical considerations, possibly combined with some calibration to market data, and typically has nice analytic representation. More recently, numerical data-driven approaches to these questions are being developed, often involving deep neural networks and machine learning techniques. In such context, the data - be it market data or generated data - is typically discrete.

In both cases above, there is fundamental uncertainty about the postulated probability measure, i.e., the model. This so-called Knightian uncertainty is of fundamental importance and a subject of intense studies in mathematics and economics alike. One way to capture the model uncertainty is through the distributionally robust approach, see [1] Distributionally robust optimization (DRO) is formulated as a mini-max problem where the inner maximization is taken over a collection of probability measures (ambiguity set) and the outer minimization is taken over all the admissible controls. The ambiguity set is often given as a small perturbation of the partially observed distributional information of the reference model.

The fundamental aim of this project is to understand both theoretical and numerical aspects of DRO problems when the ambiguity set is expressed using Wasserstein-like distances. These classical distances have recently been extended to the dynamic settings, under the name of adapted-Wasserstein metric. To define the adapted Wasserstein distance, we restrict ourselves to all causal couplings in the sense that the target process at time t only depends on the source process up to time t. This restriction makes the adapted Wasserstein distance essentially different from the classical Wasserstein distance. The new distances allow us to capture simultaneously the relevance of the information flow and of the geometry of the state space. Recent seminal results [2] show that the topology generated by the adapted Wasserstein distance agrees with other notions of adapted topology, e.g., the weak nested topology, Hellwig's information topology, Aldous' extended weak topology. And, indeed, it is the coarsest topology on the probability measure space that makes optimal stopping problems continuous. On the other hand, adapted Wasserstein distances allows us to treat discrete and diffuse measure at the same time. They also, crucially, allow us to capture the geometry of the state space, although the geodesic nature of the space of processes endowed with the adapted Wasserstein distance is still an open problem.

The project aims to consider both discrete and continuous time, as well as limiting passage from one to the other. Likewise, the aim is both to shed understanding on the DRO problem through its analysis, including duality, as well as to develop first order approximation to the value function and optimal control. This builds on the works in a one-period setting which used regular Wasserstein distances, see [3]. The DRO setting can be potentially extended to optimal stopping problems, multi-period games, risk-averse stochastic programming, etc. Applications in machine learning, mathematical finance, and statistics will be considered.

References:
[1] https://doi.org/10.1287/moor.2018.0936.
[2] https://doi.org/10.1007/s00440-020-00993-8.
[3] https://doi.org/10.1098/rspa.2021.0176.

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