Classical numerical methods for solving these equations are at the core of a key component in engineering design and simulation. Randomised versions of these schemes can offer significant improvement where the coeffients are time-irregular. In particular, Kruse and Wu have opened a new direction—using linear functional of coefficients as a superior replacement to point evaluations for solution simulation. The existing literature considers randomised Euler/Runge-Kunta method for ODEs, drift-randomised Milstein method for SDEs and fully-randomised Galerkin method for SDEs with additive noise. These randomised methods can improve the order of convergence while relaxing conditions imposed on coefficient functions.

To further develop the randomised numerical method to a reliable game chancer within the realm of financial industry and engineering, the project, led by Dr Yue Wu and Professor Xuerong Mao FRSE, will reconsider the fundamentals of functional computations. And it will address one or more of the following specific topics depending on the interest of the PhD student:

• in finance, a few products are modelled through nonlinear SDEs, for example, the interest rate. We will modify the existing method to produce an efficient randomised scheme that handles nonlinear and time-irregular SDEs, thus enable an affordable, fast, and accurate solution for real-application modelling and simulation
• SDEs with regular conditions are commonly used to model the evolving stock prices. The classical Milstein method is the popular choice in simulating such SDE solutions with its best order of convergence being one. As its advanced version, the optimal order for drift-randomised Milstein method is also limited to one. We will propose a suitable fully-randomised Milstein method for SDEs, through properly randomising the stochastic integral part, such that the optimal order of convergence could exceed one. Furthermore, we will generalise the full-randomised Milstein method to fully-randomised 2nd order method for CDEs
• establishing the Python package for randomised numerical schemes; considering the application of the randomised method on neural ordinary/controlled differential equations

The project lies at the intersection of stochastic analysis and numerical analysis, and a good background in mathematics is required. In particular, the PhD student should be comfortable with undergraduate level numerical analysis, functional analysis and probability theory.

During the PhD study, the student will receive comprehensive training and research coaching through regular one-to-one meetings with Dr Yue Wu and Professor Xuerong Mao FRSE. Plus, there will be regular research visits to DataSig team (led by Professor Terry Lyons FRS) in the University of Oxford. Such interactive and tailored support can develop their strength and consolidate their research knowledge. The PhD student will benefit from the supervisory team’s international collaborations in Europe (Germany) and Asia (China) and is expected to visit some of the international collaborators. There are dedicated support infrastructures for postgraduate students in the department and at the University of Strathclyde. And the PhD student will receive strong scientific and pastoral support in a diverse and inclusive postgraduate community.

Further information

The PhD post will commence in October 2022.