Seminar| Institute of Mathematical Sciences
Time: Wednesday, December 13th, 2023 , 14:30-15:30
Speaker: Siran Li, Shanghai JiaoTong University
Abstract: The rough path theory provides a far-reaching generalisation to the well-posedness theory for controlled/stochastic differential equations. Various fundamental problems pertaining to the signature of rough paths, which is a non-commutative analogue of polynomials and whose expectation plays a role similar to that of the moment generating function, remain widely open despite the exciting current developments in rough path analysis. In this talk, we present our recent works on the expected signature of physical Brownian motion in magnetic fields and mathematical Brownian motion stopped upon its first exit time on $C^{2,\alpha}$-Euclidean domains. We shall also briefly discuss our development of a GAN model in machine learning, known as the PCF-GAN (PCF for probability characteristic function), based on theoretical studies of the signature. Throughout these works, we emphasise a novel approach via the analysis of a graded system of parabolic PDEs in the tensor algebra. (*Joint work with Prof. Hao Ni from University College London and Qianyu Zhu from MIT).