Stochastic Differential Equations for Systems with Extreme Events

A great many real-world phenomena which involve some degree of randomness can be described with stochastic differential equations (SDEs). For instance, tumour growth or stock prices are often studied in this way [1,2]. It is very common, in models of such systems, to assume that the noise is Gaussian. However, the reality may sometimes be quite different, in that “extreme events” can occur in the underlying stochastic processes at play.

In the wake of the 2008 financial crisis, a keen interest in extreme events has arisen. These are described as either “black swans” or “dragon kings”, according to whether they are, in principle, predictable [3]. A key question is which microscopic mechanisms lead to different kinds of extreme events. For instance, “self-organised bistability” has recently been proposed as a mechanism which produces extreme events with much higher likelihood than would ensue from the power laws associated with self-organised criticality [4].

It is possible that many large-scale (i.e. macroscopic) extreme events - such as stock-market crashes, pandemics, wars or even the metastasis of a tumour - may be caused by microscopic extreme events in the underlying stochastic processes. If so, these phenomena might be described by considering the correct noise terms in the associated SDEs. Furthermore, the advent of big data and machine learning methods have recently opened the door to analysing such processes empirically. The aim of this project is to study how microscopic extreme events in the noise terms of SDEs can lead to macroscopic extreme events; to obtain and analyse data from finance, biology and other fields in search of signatures of such phenomena; and to develop methods for solving SDEs of this kind.

A possible new application in the field of cancer is the investigation of extreme mechanical properties, experimentally observed by Plodinec et al [5], at the University of Basel. Here, the distributions of stiffness in breast cancer is characterised by an increasing mean and standard deviation of cell stiffness during initial tumour progression; while advanced metastatic tumours exhibit multi-modal distributions. Their original work ignored the heavy tails in such distributions which can be observed in their data, and which may, therefore, play a decisive role in metastasis. Dr Spill’s preliminary work (in revision at Science Advances) shows that extremal mechanical properties may induce malignant phenotypes in vitro, and we are in contact with the group in Basel to develop a collaboration to infer the role of extreme mechanical properties in cancer patients. We are also in contact with AstraZeneca in Cambridge to obtain data about potential treatments that target mechanical signatures of cancer.

The student would also benefit from interaction with the group of Prof Miguel Ángel Muñoz, at the University of Granada, who have expertise in the physics of extreme events, and already collaborate closely with Dr Johnson [4,6]. We would seek to build collaborations at the Alan Turing Institute with the “Extreme event prediction and monitoring” research project, led by Prof Axel Gandy and Prof Almut Veraart; and with the "Population diversity at varying scales" project, led by Prof Jean-Baptiste Cazier.

Given its intended application to cancer, this project addresses mainly the Challenge “Revolutionise healthcare”. However, the methods we will develop will be useful for modelling and understanding other extreme events, and hence the work can also be expected to contribute both to “Shine a light on our economy” and “Manage security in an insecure world”.

Required:
A strong background in statistics and either financial engineering or mathematical biology. Moreover, the candidate should have a strong programming background in Python, R or C++.

Desired:
Some experience in analysis of rare event statistics and stochastic processes. Programming experience specifically regarding data analysis or model development in finance, biology or medicine.