## Summary

## Research

Professor Thomas Cass is on sabbatical leave at the Institute for Advanced Study (IAS) in Princeton, NJ for the academic year 2023-24.

He is Professor at the Mathematics Department at Imperial College London and a Visiting Researcher at the Alan Turing Institute in London. He completed his MA and PhD at the University of Cambridge, and he has previously held positions at the University of Oxford. His research interests span a number of areas of pure and applied mathematics; he has written fundamental research papers in the areas of probability, stochastic analysis as well topics in mathematical data science and in mathematical finance. Much of his research uses the theory of rough paths and rough analysis, which has rapidly become a major tool in mathematics over the past two decades. Its applications can now be across multiple research areas, including probability and analysis, mathematical finance and, mostly recently, in data science.

## Projects and Leadership Roles

Thomas is involved in major long-term projects funded by the UK Research and Innovation (UKRI). He is a Co-Investigator on the EPSRC Programme Grant EP/S026347/1, **Unparameterised multi-modal data, high order signatures, and the mathematics of data science (DataSig) .**

This five-year project (2019-2024) will embed rough path analysis as a core tool in the data science of multimodal unparameterized streams, and will develop broad computational tools derived from four challenge areas: computer vision, radioastronomy, human-computer interaction and mental health. More details on the DataSig project and its activities and outputs can be found on the DataSig website.

Thomas is a Co-Director, and Imperial College lead, for **the EPSRC Centre for Doctoral Training (CDT) in the Mathematics of Random Systems: Analysis, Modelling and Simulation**, a collaboration between Imperial College and the University of Oxford.

The CDT is training the next generation of inter-disciplinary researchers across a spectrum of research areas in stochastic analysis, modelling and data science. Information on the CDT, its activities, industry sponsors and the current student cohorts can be found on the CDT's website.

## Selected Publications

### Journal Articles

Cass T, Turner W, 2024, Topologies on unparameterised path space, *Journal of Functional Analysis*, Vol:286, ISSN:0022-1236

Cass T, Lyons T, Xu X, 2024, Weighted signature kernels, *Annals of Applied Probability*, Vol:34, ISSN:1050-5164, Pages:585-626

Cass T, Pei J, 2023, A Fubini type theorem for rough integration, *Revista Matematica Iberoamericana*, Vol:39, ISSN:0213-2230, Pages:761-802

Cass T, Ferrucci E, 2023, On the Wiener chaos expansion of thesignature of a Gaussian process, *Probability Theory and Related Fields*, ISSN:0178-8051

Cass T, Messadene R, Turner WF, 2023, Signature asymptotics, empirical processes, and optimal transport, *Electronic Journal of Probability*, Vol:28, ISSN:1083-6489, Pages:1-29

Armstrong J, Brigo D, Cass T, et al. , 2022, Non-geometric rough paths on manifolds, *Journal of the London Mathematical Society - Second Series*, Vol:106, ISSN:0024-6107, Pages:756-817

Salvi C, Cass T, Foster J, et al. , 2021, The signature kernel is the solution of a Goursat PDE, *Siam Journal on Mathematics of Data Science*, Vol:3, ISSN:2577-0187, Pages:873-899

Armstrong J, Bellani C, Brigo D, et al. , 2021, Option pricing models without probability: a rough paths approach, *Mathematical Finance*, ISSN:0960-1627

Cass T, Crisan D, Dobson P, et al. , 2021, Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes, *Electronic Journal of Probability*, Vol:26, ISSN:1083-6489, Pages:1-72

Cass T, Lim N, 2021, Skorohod and rough integration for stochastic differential equations driven by Volterra processes, *l' Institut Henri Poincare. Annales (B). Probabilites et Statistiques*, Vol:57, ISSN:0246-0203, Pages:132-168

Bensoussan A, Cass T, Chau MHM, et al. , 2020, Mean field games with parametrized followers, *IEEE Transactions on Automatic Control*, Vol:65, ISSN:0018-9286, Pages:12-27

Cass T, Lim N, 2019, A Stratonovich-Skorohod integral formula for Gaussian rough paths, *Annals of Probability*, Vol:47, ISSN:0091-1798, Pages:1-60

Cass T, Ogrodnik M, 2017, Tail estimates for Markovian rough paths, *Annals of Probability*, Vol:45, ISSN:0091-1798, Pages:2477-2504

Cass T, Driver BK, Lim N, et al. , 2016, On the integration of weakly geometric rough paths, *Journal of the Mathematical Society of Japan*, Vol:68, ISSN:0025-5645, Pages:1505-1524

Cass T, Driver BK, Litterer C, 2015, Constrained Rough Paths, *Proceedings of the London Mathematical Society*, Vol:111, ISSN:1460-244X, Pages:1471-1518

Cass T, Lyons T, 2015, Evolving communities with individual preferences, *Proceedings of the London Mathematical Society*, Vol:110, ISSN:0024-6115, Pages:83-107

Cass T, Clark M, Crisan D, 2014, The filtering equations revisited, *Springer Proceedings in Mathematics and Statistics*, Vol:100, ISSN:2194-1009, Pages:129-162

Cass T, Litterer C, Lyons T, 2013, Integrability and tail estimates for Gaussian rough differential equations, *Annals of Probability*, Vol:41, Pages:3026-3050

Cass T, Hairer M, Litterer C, et al. , 2012, Smoothness of the density for solutions to Gaussian Rough Differential Equations

Cass T, Litterer C, 2011, On the error estimate for cubature on Wiener space

Cass T, Friz P, 2011, Malliavin calculus and rough paths, *Bulletin Des Sciences Mathematiques*, Vol:6-7, Pages:542-556

Cass T, Friz, P., 2010, Densities for rough differential equations under Hörmander’s condition, *Annals of Mathematics*, Vol:171, ISSN:0003-486X, Pages:2115-2141

Cass T, 2009, Smooth densities for solutions to stochastic differential equations with jumps, *Stochastic Processes and Their Applications*, Vol:119

Cass T, Friz P, Victoir N, 2009, Non-degeneracy of Wiener functionals arising from rough differential equations, *Transactions of the American Mathematical Society*, Pages:3359-3371