Get to know Dr Andy Allan from our Department of Mathematical Sciences.
My name is Andy, and I grew up in Biggin Hill, a small town in South East England. After leaving school I went to study mathematics at Oxford University, where I then became a DPhil student in the Mathematical and Computational Finance group. I’ve always enjoyed probability, differential equations and mathematical analysis, and it was therefore natural for me to end up working in the intersection of these fields, namely stochastic analysis, which, broadly speaking, is the study of the dynamics of randomly evolving systems. After graduating in the summer of 2019, and before moving to Durham, I lived in Switzerland for 3 years, working as a postdoctoral researcher in the Stochastic Finance group at ETH Zurich.
I joined Durham in September 2022 as an assistant professor in the Probability group in the Department of Mathematical Sciences. Alongside my research, supervision and tutorial teaching, I currently teach a lecture course on financial mathematics for master’s students.
My current research revolves around the theory of rough paths. This is a recent branch of stochastic analysis, designed to study nonlinear interactions between highly oscillatory (i.e., “rough”) signals. Much of my recent work has been in developing a framework for rough paths adapted for applications to financial mathematics. I’m also currently working on the convergence of numerical schemes for differential equations driven by such rough paths.
The theory of rough paths focuses on a particular characteristic of a path (i.e., an evolving system or data stream), known as its signature, which is the collection of all the iterated integrals of the path against itself. It can be shown that the signature encodes precisely the information required to define a continuous solution map for differential equations driven by paths which are too irregular to be handled by classical calculus.
This provides a completely new approach to analyse stochastic processes, which doesn’t rely on the specification of a probabilistic model, allowing for the derivation of statistics and strategies which are robust to model uncertainty. Moreover, the signature turns out to possess many desirable properties which make it suitable as a feature map for streamed data. For instance, nonlinear functions on paths become approximately linear functions on the signature, and a plethora of machine learning algorithms based on this “universal nonlinearity” are now being developed.
One of the main applications of my research is to stochastic filtering, which is the problem of determining the posterior distribution of a hidden process as it evolves randomly in time from noisy data, a problem which arises in various fields, particularly in finance and engineering. My research concerns constructing stochastic filters which are robust, both with respect to uncertainty in the model parameters, and with respect to errors in the observed data stream. It turns out that rough path theory is an ideal tool for this problem, due to its strong continuity results for nonlinear stochastic systems. Another application of my research is to mathematical finance, where the rough path perspective has implications for the construction of financial portfolios which are robust with respect to model uncertainty.
Some systems involve two or more signals which affect the system in different ways, and it can be helpful to condition on one signal, whilst treating the others as stochastic (i.e., random) processes. A natural approach to the analysis of such systems leads to hybrid differential equations, which are driven by both rough and stochastic noise. The interaction between these two distinct types of noise requires delicate analysis and opens up new applications in areas including stochastic filtering and systemic risk modelling.
I enjoy walking, hiking and traveling, and can usually be found listening to a podcast or audiobook.
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