Making Cubature on Wiener Space Work

Quadrature (or in higher dimensions cubature) is a classical method for calculating areas and historically related to the development of the integral calculus. In its modern form it goes back to the work of Gauss and refers to the approximation of the definite integral of a function by a weighted sum of function values at a finite number of carefully chosen points. The work of Lyons and Victoir has combined this fundamental idea with the machinery of modern stochastic analysis and applied it on infinite dimensional path spaces. This has resulted in a novel particle method that can be used to track the evolution of a large class of random systems. The approximation convergences rapidly and is robust as the particles evolve unlike in classical methods (Euler) along admissible trajectories. Moreover, while the underlying ideas are probabilistic the approximation is deterministic.

In filtering problems, we aim to make reasonable inferences about the evolution of complex phenomena based on partial observations of the system. Such problems are natural and come in virtually all shapes and sizes: from the focus of a camera in a mobile tracking a moving object, via the imaging produced by a modern MRI scanner in hospital, to the prediction of next week's weather by means of a supercomputer. The aim of the proposed research is to help to transform cubature on Wiener space from a promising and novel approach to numerical integration "in the lab" to a powerful method that can easily be adopted by practitioners to help solve such problems that impact our lives. The proposed research will bring together ideas from probability, numerical analysis and algebra to gain a more systematic understanding of the construction of cubatures on path space. These cubatures result in highly efficient particle methods that combine rapid convergence with transparent bounds on the complexity of the particle descriptions of the evolving measures. As part of this project we want to lower the hurdle for other researchers working in academia and industry to adopt our ideas. Hence, we propose to develop efficient and accessible C++ implementations of the numerical methods and to contribute them to the existing open source computational rough path library.

We have identified five potential areas for societal and economic impact.

1) Applications to the efficient pricing of financial derivatives
Financial services are a key industry in the United Kingdom contributing in 2018 more than £132 billion to the UK economy and accounting for 6.9% of the total economic output [2]. The notional volume of Euro denominated derivative contracts (including options, swaps, futures) cleared in London can exceed £900 billion in a single day. Partial differential equations routinely arise when pricing derivatives at the trading desks of investment banks or energy companies. Two key requirements for PDE based methods are speed/memory consumption and the consistent repricing of calibration instruments (typically linear instruments and options). Constructing high-order cubature formulas on Wiener space and enhanced recombination algorithms proposed in this project leads to very efficient numerical methods capable of computing high accuracy PDE solutions. Our algorithms are in moderate dimensions an excellent match for these requirements.

Financial examples will lead to well tested software and concrete innovations that can in turn be intermediate steps to solving general PDE based problems. Increased numerical efficiency gives rise to broader societal benefits lowering the amount of computational resources and energy required to solve these problems.

2) Applications to data-assimilation , non-linear filtering and methods for solving high-order PDE
Non-linear filtering problems arise in areas as diverse as earth sciences (climate modelling and oceanography [1]), robotics (e.g. compensation for tremors in surgical application [3]), defense (multi-target tracking problems arising in ballistic missile defense [4]). Beyond the obvious economic benefits, solving these increasingly complex problems efficiently and accurately is a crucial challenge that helps to inform public decision making. In Objective 3.1 we propose to develop a cubature-based particle approximation to the non-linear filtering problem that converges more quickly than existing methods with improved computational efficiency.

New approximations to high-order PDE solutions can benefit the work of applied scientists and engineers in areas that rely on geometric design. Here, high-order PDEs arise for example in the context of medical imaging or modelling wound geometry.

3) Computing invariant functionals of the signature and applications in data-science
In Objective 3.3 we propose to construct Caratheodory cubature measure to efficiently compute (rotation) invariant functionals of the signature. These cubatures can benefit machine learning applications based on the signatures of sequential data streams. Examples of such applications include Chinese handwriting recognition and classification problems for medical data and images.

4)Training of PhD students
The availability of post-graduates with deep knowledge of both the analytic and computational aspects of cutting-edge numerical methods is essential for maintaining London's role as a leading centre for quantitative finance. As part of a successful grant application the Department of Mathematics at the University of York will commit a fully funded PhD studentship.

5)Impact on teaching

References:
[1] Evensen, G. Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. of Geophysical Res.: Oceans (1994)

[2] House of Commons Research Briefing: Financial services: contribution to the UK economy.

[3] Veluvolu, K. C. et al. Estimation and filtering of physiological tremor for real-time compensation in surgical robotics applications. Int. J. of Medical Robotics and Computer Assisted Surgery 6.3 (2010): 334-342.

[4] Ristic, B., et al. Performance bounds and comparison of nonlinear filters for tracking a ballistic object on re-entry. IEE Proc.-Radar, Sonar and Navigation 150(2003): 65-70