Alex Watson

2024/25, Term 1: Optimisation and Operations Research

For previous teaching, please see the Teaching Archive.

The UCL probability group: Probability at UCL

I am currently seeking candidates for the following PhD project:

Branching processes

Branching processes are a topic in probability theory, offering a model of a collection of "cells" which change and divide. Each cell has some associated trait, such as mass. As time passes, the trait changes (e.g., the cell grows) and eventually the cell divides into two new cells, which have the same behaviour.

Models like this are widely used in biology and physics, but they also have significant theoretical interest, which is the focus of this project. One interesting property is that as time progresses, the cell trait distribution typically settles down into a steady state, which can be described. Another is that as the initial number of cells is made large, the random evolution of the model reduces to a deterministic one, possibly with a random component. The project is to investigate these "asymptotic" (large time or large cell count) properties. When do they hold, and when not? How can we describe the "limit" objects? And how does the presence of interactions, such as direct or indirect competition, change the situation?

As this project concerns probability theory, you should have a solid mathematical background with a good understanding of stochastic processes (particularly Markov processes), measure theory and martingales.

If you are interested, please get in touch via email (contact details below).

Research students

• Ramkrishna Samanta

Graduated PhD students

• Xin Xu (with Kees van Schaik): Applications of Lévy processes in insurance with omega killing
• Dalal Alghanim (with Ronnie Loeffen): Optimal control of taxation for spectrally negative Lévy processes

• W. Da Silva, E. Powell, A. R. Watson. Growth-fragmentations, Brownian cone excursions and SLE(6) explorations of a quantum disc. 2025.
  Preprint
• D. Al Ghanim, R. Loeffen, A. R. Watson. Stochastic optimal control of Lévy tax processes with bailouts. 2024.
  Preprint
• D. Villemonais, A. R. Watson. A quasi-stationary approach to the long-term asymptotics of the growth-fragmentation equation. Ann. Appl. Probab., 2025.
  DOI PDF Preprint
• K. van Schaik, A. R. Watson, X. Xu. Optimal stopping of the stable process with state-dependent killing. Bernoulli, 2025.
  DOI Preprint
• L. Döring, L. Trottner, A. R. Watson. Markov additive friendships. Trans. Amer. Math. Soc., 2024.
  DOI Preprint
• L. Döring, M. Savov, L. Trottner, A. R. Watson. The uniqueness of the Wiener-Hopf factorisation of Lévy processes and random walks. Bull. Lond. Math. Soc., 2024.
  DOI Preprint
• A. R. Watson. A growth-fragmentation connected to the ricocheted stable process. J. Appl. Probab., 2022.
  DOI Preprint
• E. Horton, A. R. Watson. Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes. ALEA, 2022.
  DOI Preprint
• J. Bertoin, A. R. Watson. The strong Malthusian behavior of growth-fragmentation processes. Ann. H. Lebesgue, 2020.
  DOI Preprint
• D. Al Ghanim, R. Loeffen, A. R. Watson. The equivalence of two tax processes. Insurance Math. Econom., 2020.
  DOI Preprint
• Q. Shi, A. R. Watson. Probability tilting of compensated fragmentations. Electron. J. Probab., 2019.
  DOI Preprint
• L. Döring, A. R. Watson, P. Weißmann. Lévy processes with finite variance conditioned to avoid an interval. Electron. J. Probab., 2019.
  DOI Preprint
• J. Bertoin, A. R. Watson. A probabilistic approach to spectral analysis of growth-fragmentation equations. J. Funct. Anal., 2018.
  DOI Preprint
• J. Bertoin, A. R. Watson. Probabilistic aspects of critical growth-fragmentation equations. Adv. in Appl. Probab., 2016.
  DOI Preprint
• A. E. Kyprianou, A. R. Watson. Potentials of stable processes. Séminaire de Probabilités, 2014.
  DOI Preprint
• A. E. Kyprianou, J. C. Pardo, A. R. Watson. The extended hypergeometric class of Lévy processes. J. Appl. Probab., 2014.
  DOI Preprint
• A. Kuznetsov, A. E. Kyprianou, J. C. Pardo, A. R. Watson. The hitting time of zero for a stable process. Electron. J. Probab., 2014.
  DOI Preprint
• A. E. Kyprianou, J. C. Pardo, A. R. Watson. Hitting distributions of alpha-stable processes via path censoring and self-similarity. Ann. Probab., 2014.
  DOI PDF Preprint HTML LaTeX
• A. R. Watson. Stable processes. PhD thesis, University of Bath, 2013.
  Link

• A growth-fragmentation found in the cone excursions of Brownian motion (and in the quantum disc)
  16 October 2024
  Isaac Newton Institute seminar
  Slides HTML LaTeX
• Uniqueness of the Wiener-Hopf factorisation of Lévy processes
  22 April 2024
  King's College London Probability Seminar
  Slides HTML LaTeX
• Markov additive friendships
  18 July 2022
  10th International Conference on Lévy Processes
  Slides HTML
• Growth-fragmentation and quasi-stationary methods
  17 September 2021
  Bath-Beijing-Paris Branching Structures/Neutron Transport Days
  Slides Video HTML LaTeX
• Strong laws for growth-fragmentation processes with bounded cell size
  6 April 2021
  IECL seminar
  Slides HTML LaTeX
• Risk processes with tax
  17 July 2020
  UCL Statistical Science Seminar
  Slides HTML
• Self-similar growth-fragmentation models
  8 November 2016
  BIRS-CMO Workshop on Stable Processes
  Video

UK Easter Probability Meeting · Manchester, 27 – 31 March 2023

Dimers and related models
Online, 22 September 2021

Rough path theory in machine learning
Online, 26 August 2020

UK Easter Probability Meeting
Manchester, 30 March - 3 April 2020 (postponed)

Growth and division in mathematics and medicine
University College London, 4 - 6 Nov 2019

Growth-fragmentation phenomena
Royal Statistical Society, 12 Sep 2019

Stochastic processes in finance and beyond
ICMS Edinburgh, 23 May 2019

PiNE in Manchester
Manchester, 12 September 2018

Meeting on Lévy processes and anomalous diffusion
Manchester, June 2017

Soapbox Science Manchester
Manchester, July 2016

Zürich Spring School on Lévy Processes
Zürich, Spring 2015

Junior multiscale mechanics and probability meeting
Bath, December 2011

• Email

  alexander.watson@ucl.ac.uk

• Location

  Correspondence: Department of Statistical Science, University College, Gower Street, London, WC1E 6BT, UK. Location: Office 135a, 1st floor, 1-19 Torrington Place, London, WC1E 7HB

• Office hours: check Moodle for taught courses, personal tutees by appointment

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