Mathematics of Random Systems Summer School 2022
Sunday 25th - Friday 30th September St Hugh's College, St Margaret's Rd, Oxford, OX2 6LE, UK
The 2022 Summer School in Mathematics of Random Systems is jointly organised by the EPSRC CDT in Mathematics of Random Systems and the Berlin-Oxford International Research Training Group (IRTG) Stochastic Analysis in Interaction. The school will feature courses on topics of current interest in Stochastic Analysis as well as presentations by students from both Oxford and Berlin.
Registration to join the summer school has now closed.
Download the programme
There are a limited number of bursaries available for students working on related topics to cover the registration costs. To apply for these please send your CV and letter of support from you supervisor to melanie.witt@maths.ox.ac.uk.
Photo Akshunna S. Dogra
Stochastic Quantisation
In this minicourse I will give an overview on recent results and techniques that allow the use of stochastic analysis to study certain measures on the spaces of distributions over two and three dimensional Euclidean space which are usually known as Euclidean quantum fields. The analysis of such measures is plagued by both small scale and large scale singularities. By following basic ideas of stochastic analysis one can identify suitable building blocks and reasonably simple equations which allows the construction of such measures. This program goes under the generic name of "stochastic quantisation". We will cover basic ideas of stochastic quantisation and the relation between the properties of the measures in relation to its stochastic quantisation, e.g.: existence, uniqueness, cluster properties, etc.
References
- Gubinelli, M. and Hofmanova, M (2021) A PDE Construction of the Euclidean $\Phi^4_3$ Quantum Field Theory, ArXiv:1810.01700
- Barashkov, N. and Gubinelli, M. (2020) A Variational Method for $\Phi^4_3$, Duke Mathematical Journal, 169(17): 3339–3415
Additional resources:
Optimal transport theory and Wasserstein distances
The aim of the minicourse is to present recent advances related to optimal transport and its applications in mathematical finance, statistics, optimization and beyond. We will discuss basics of optimal transport (OT), its duality theory and properties of the induced Wasserstein distance on the space of probability measures. I will then introduce the martingale version of the problem (MOT) and discuss the rich additional structure resulting from the martingale constraint. I will touch on numerics for both problems, including the entropic relaxation of the OT. Finally, I will discuss how Wasserstein distances can be used to develop robust data-driven approaches to modelling in mathematical finance, machine learning and beyond.
References:
- C Villani (2009) Optimal Transport: Springer:
- Filippo Santambrogio (2015), Optimal Transport for Applied Mathematicians, Springer
- Cedric Villani (2003), Topics in Optimal Transportation, AMS
MOT basics:
There are several papers here and books but there is no single basic reference work I would recommend.
Instead, maybe it is helpful to point to a variety of papers which showcase the relevant topics (not all will be covered though):
- Beiglböck, M., Henry-Labordère, P. & Penkner, F. Model-independent bounds for option prices—a mass transport approach. Finance Stoch 17, 477–501 (2013)
- Beiglböck, M., Cox, A.M.G. & Huesmann, M. Optimal transport and Skorokhod embedding. Invent. math. 208, 327–400 (2017)
- Guo, G. & Obłój, J. Computational methods for martingale optimal transport problems. Ann. Appl. Probab. 29 (6), 3311 - 3347, (2019)
- Oblój, J & Wiesel, J. Robust estimation of superhedging prices. Ann. Statist. 49(1) 508 - 530, (2021)
- Bartl, D. Drapeau, S. Oblój, J. and Wiesel, J. Sensitivity analysis of Wasserstein distributionally robust optimization problems. Proc. R. Soc. A. 477 20210176 (2021)
