For Immediate Release – 17 June 2025
MUT staff member achieves a rare feat in his Maths PhD study
Achieving a PhD in Mathematics is still a big thing. Mathematics is regarded as a “difficult’ subject. Having said that, MUT has three staff members with PhDs in this “difficult” subject. This is not new at the University; two Maths PhDs have recently retired. Among the three that are still in the system, one is a Dean of the Faculty of Applied and Health Sciences; one is a Head of the Department of Mathematical Sciences. The latest is the most interesting case. Darlington Hove, a Lecturer in the department, has just obtained his PhD from the University of Limpopo. Dr Hove’s topic was “Local times of deterministic paths and model-free cádlág price paths and their application to mathematical finance”. This is a real tongue twister. Cádlág, French word which means “right-continuous with left limits. This is sometimes abbreviated as “RCLL”. The topic Dr Hove chose is Applied Mathematics. Most people would say that it is very rare to have someone obtaining a PhD in Applied Mathematics.
Dr Hove’s research explored the concept of local times for deterministic and cádlág model-free paths, building on Hans Föllmer’s pioneering pathwise approach to Itô calculus. By utilising the notion of truncated variation, we establish the weak convergence of normalised interval crossing numbers and leverage this framework to construct illustrative examples on local times. During the process of his study, Dr Hove proved a new mathematical result – a theorem that links local time to how frequently a price path crosses certain levels for functions with finite variation.
Dr Hove responded to a call for PhD funding in various areas of Applied Mathematics, offered under the SA-UK University Staff Development Programme (USDP). “My application was successful, and I joined a cohort of eight other PhD Applied Mathematics students studying under different concentrations. My primary interest was in Financial Mathematics, and I was particularly drawn to one of the available research topics listed under an umbrella project in this area.”
Prior to sending in his applying, Dr Hove had already been conducting independent research in Financial Mathematics, having decided that this would be the focus of his doctoral studies. The proposed topic captured his attention as it focused on the local times of cádlág security price paths, a niche and under-researched area within stochastic processes and quantitative finance.
Dr Hove gives more reasons for the selecting this area of study. “Moreover, the opportunity to work under the supervision of a researcher actively developing novel approaches to the treatment of local times further motivated my decision.” Dr Hove said that a combination of a relevant research interest, a unique and underexplored topic, and a supportive supervisory environment aligned well with both his academic goals and his long-term research aspirations.
Dr Hove’s thesis explores how financial market prices can be studied without relying on traditional mathematical models. “The main reason for moving away from these traditional models is that they have often failed to capture the full complexity and interconnectedness of the financial system. This shortcoming contributed to the 2008 global financial crisis, which nearly brought the global financial system to a standstill,” he said.
In his research, Dr Hove used a model-free approach, which does not rely on strict assumptions that are in the well-known Black-Scholes-Merton model. “Instead, it only requires one basic condition: there should be no way to make unlimited profit from a very small investment, a fair principle in well-functioning markets,” he said.
Dr Hove’s focus was on a concept called local time, which helps researchers to understand how long a price stays at a certain level. “I used a mathematical method known as Föllmer’s pathwise Itô calculus to show that if local time is missing from a price path, it may allow someone to make extremely large profits simply by timing their trades well—something that should not happen in a fair market,” he said.
Dr Hove explains further how he achieved this very feat. “To explore this further, I used a method called truncated variation, developed by my supervisor, Professor Rafal Łochowski, to study how often prices cross certain levels. I also created examples to show when local time exists and when it doesn’t. One important result was constructing a path (inspired by the Cantor set) which has quadratic variation but lacks local time – and then modifying it to bring local time into existence.”
Dr Hove also explored I also explored Peano curves, which are complicated paths that typically fill unit 2D space using unit segments – and developed an algorithm to calculate how often these paths cross intervals. This helped determine the conditions for when the local time is present in such paths.
Dr Hove graduated on 12 April 2025.
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