Thomas's research focuses around stochastic analysis, especially rough path theory and its applications to problems in the stochastic processes, stochastic differential geometry and its interaction with Malliavin's calculus.
Rough path theory emerged as a branch of stochastic analysis with the aim of giving an improved approach to dealing with the interactions of complex random systems. It has since matured in a number of exciting directions of both theoretical and practical interest. At the core of rough path theory is the signature transform which, while being simple to define, has rich mathematical properties bringing in aspects of analysis, geometry and algebra. It allows on to define the space of functions on unparameterised paths/data stream in a very economical way. The past 5 years a significant strand of applied work has been undertaken to exploit the mathematical richness of this object in diverse data science challenges.
- Thomas Cass and Nengli Lim, Skorohod and rough integration for stochastic differential equations driven by Volterra processes. Accepted (2020), to appear Annales de l'Institut Henri Poincare (B)
- Alain Bensoussan, Thomas Cass, Man Ho Michael Chau, and Sheung Chi Phillip Yam. Mean Field Games With Parametrized Followers IEEE Transactions on Automatic Control 65, no. 1 (2019): 12-27.
- Thomas Cass and Nengli Lim A Stratonovich-Skorohod integral formula for Gaussian rough paths Ann. Probab. Volume 47, Number 1 (2019), 1-60.
- Thomas Cass and Marcel Ogrodnik. Tail estimates for Markovian rough paths Ann. Probab., Volume 45, Number 4 (2017), 2477-2504.
- Thomas Cass, Bruce K. Driver, Nengli Lim, Christian Litterer. On the integration of weakly geometric rough paths, J. Math. Soc. Japan, Volume 68, Number 4 (2016), 1505-1524.
- Thomas Cass, Bruce K. Driver and Christian Litterer. Constrained rough paths, Proc. London Math. Soc. (2015) 111 (6): 1471-1518.
- Thomas Cass, Martin Hairer, Christian Litterer, and Samy Tindel. Smoothness of the density of solutions to Gaussian rough differential equations, Ann. Probab., Vol. 43 (2015), No. 1, 188–239
- Thomas Cass and Christian Litterer. A gradient estimate for the heat semi-group without hypoellipticity assumptions, Proc. Amer. Math. Soc. 143 (2015), no. 11, 4967–4972.
- Thomas Cass and Terry Lyons. Evolving communities with individual preferences, Proc. London Math. Soc., (3) 110 (2015) no.1, 83-107.
- Thomas Cass, Dan Crisan, Martin Clark. The filtering equations revisited. In: Stochastic Analysis and Applications 2014 Springer Proceedings in Mathematics & Statistics, Vol. 100, (Crisan, Hambly and Zariphopoulou eds.) in honour of Terry Lyons.
- Thomas Cass and Christian Litterer. On the error estimate for cubature on Wiener space. Proc.Edin. Math. Soc., (Series 2), Vol. 57 no.2, (2014) 377-391
- Thomas Cass, Christian Litterer, and Terry Lyons. Integrability and tail estimates for Gaussian rough differential equations, Ann. Probab. 41 (2013), no. 4, 3026-3050
- Thomas Cass, Zhongmin Qian, and Jan Tudor, Non-linear evolution equations driven by rough paths. Stochastic analysis and applications to finance, 1–18, Interdiscip. Math. Sci., 13, World Sci. Publ., Hackensack, NJ, (2012)
- Thomas Cass, Christian Litterer, and Terry Lyons, Rough paths on manifolds, New trends in stochastic analysis and related topics, Interdiscip. Math. Sci., vol. 12, World Sci. Publ.,Hackensack, NJ, (2012), 33-88.
- Thomas Cass and Peter Friz: Malliavin Calculus and Rough Paths, Bull. des Sciences Math, vol. 135, issues 6-7 (2011), 542-556.
- Thomas Cass and Peter Friz:. Densities for rough differential equations under Hörmander’s condition. Ann. of Math. vol. 171 (2010) 2115-2141 Issue 3.
- Thomas Cass, Peter Friz: and Nicolas Victoir. Non-degeneracy of Wiener functionals arising from rough differential equations, Trans. Amer. Math. Soc. 361 (2009), no. 6, 3359-3371.
- Thomas Cass. Smooth densities for stochastic differential equations with jumps, Stoch. Process. Appl. (2009), no.5, 1416-1435.
- Ian Sabir, James Fraser, Thomas Cass, Andrew Grace and Chris Huang A quantitative analysis of the effect of cycle length on arrthymogenicity in hypokaleamic Langendorff-perfused murine hearts, with I. Sabir, J. Fraser, A. Grace and C. Huang, Pfulgers Arch. September (2007); 454(6); 925-36
Links to recent preprints
- Thomas Cass, Terry Lyons, Cristopher Salvi, Weixin Yang: Computing the full signature kernel as the solution of a Goursat problem https://arxiv.org/abs/2006.14794
- Thomas Cass, Goncalo dos Reis, William Salkeld, Rough functional quantization and the support of McKean-Vlasov equations https://arxiv.org/abs/1911.01992
- John Armstrong, Claudio Bellani, Damiano Brigo, Thomas Cass, Option pricing models without probability: a rough paths approach https://arxiv.org/abs/1808.09378
- Thomas Cass, Dan Crisan, Paul Dobson, Michela Ottobre: Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes https://arxiv.org/abs/1805.01350
- Thomas Cass and Martin P. Weidner: Tree algebras over topological vector spaces in rough path theory https://arxiv.org/abs/1604.07352.
- Thomas Cass and Martin P. Weidner Hörmander's theorem for rough differential equations on manifolds http://arxiv.org/abs/1604.07352.