This book presents a selection of cutting-edge contributions from leading experts in the field, capturing the latest developments in stochastic analysis and its growing interface with neighboring disciplines. Stochastic analysis is a rapidly evolving branch of mathematics focused on the behavior of dynamical systems influenced by randomness. Over the past three decades, it has grown into one of the most vibrant and interdisciplinary areas of research, with profound impact on fields ranging from finance and physics to data science and engineering.

Topics include rough path theory, stochastic control, stochastic partial differential equations, random matrices, and applications in machine learning.

Building on the success of a recent international conference that brought together researchers from both academia and industry, this proceedings book highlights the depth and breadth of current work in the field. It serves as a valuable resource not only for academic researchers in mathematics, but also for practitioners working in areas such as quantitative finance, data-driven modeling, and applied probability.

Autorentext

Dan Crisan is a professor of Mathematics at Imperial College London and Director of the Centre for Doctoral Training in the Mathematics for our Future Climate. He has a Ph.D. in Mathematics from the University of Edinburgh having graduated in December 1996 with a thesis entitled The Problem of Nonlinear Filtering. After his Ph.D. studies, he held a postdoctoral fellowship at Imperial College London and moved to the Statistical Laboratory in Cambridge as an assistant lecturer. Crisan returned to Imperial in 2000, where he was awarded a prestigious Governors' Lectureship. He was promoted to a full professor in 2011. His research is in the wider area of Stochastic Analysis and application.

Ilya Chevyrev is currently a researcher at SISSA. Prior to this, he was a Ph.D. student and junior research fellow at the University of Oxford, a postdoc and Mercator Fellow at TU Berlin, and a Reader at the University of Edinburgh. His research focuses on rough analysis.

Thomas Cass is Professor of Mathematics at Imperial College London. His research focuses on probability, stochastic analysis, and mathematical data science, with particular emphasis on rough path theory. He completed his PhD at the University of Cambridge and held academic positions at the University of Oxford before joining Imperial in 2011. At Imperial, he directs the EPSRC Centre for Doctoral Training in the Mathematics of Random Systems, a joint initiative with the University of Oxford, and is part of the leadership team for the DataSig I and DataSig II programmes, which develops mathematical tools for analysing complex data. He has been a Visiting Researcher at the Alan Turing Institute and was the Erik Ellentuck Fellow at the Institute for Advanced Study in Princeton.

James Foster is currently a lecturer in mathematics at the University of Bath. Previously, he was a Ph.D. student and postdoctoral researcher at the University of Oxford. His research focuses on stochastic numerics, differential equations and their applications to machine learning.

Christian Litterer is currently a Lecturer in the Department of Mathematics at the University of York. He completed his PhD and held a postdoctoral position at the University of Oxford. Before joining the University of York, he also held postdoctoral positions at Imperial College London and École Polytechnique. His research interests lie in signatures, rough paths, and their applications in stochastic analysis.

Cristopher Salvi is a lecturer in Mathematics and AI at Imperial College London, Department of Mathematics, and Imperial X. Previously, he was a Chapman fellow in Mathematics at Imperial College London and before that he obtained his Ph.D. from the University of Oxford under the supervision of Terry Lyons.

Inhalt

Approximating the signature of Brownian motion for high order SDE simulation.- Randomisation of rough stochastic differential equations.- A Canonical Signature-Based Feature Set for Multivariate Time Series Classification.- Twin Brownian particle method for the study of Oberbeck-Boussinesq fluid flows.- Lower Bounds for the Support of Cubature Measures on Wiener Space and Optimal Degree-five Constructions.- Free Groups and Signatures.- A representation for the Expected Signature of Brownian motion up to the first exit time of the planar unit disc.- Asymptotic expansions of central limit distances in Vaserstein metrics.- A neural RDE-based model for solving path-dependent parabolic PDEs.- A Data-driven Market Simulator for Small Data Environments.- Mimicking and Conditional Control with Hard Killing.- Continuous random field solutions to parabolic SPDEs on p.c.f. fractals.- Pricing American options under rough volatility using signatures.- A new architecture of high-order deep neural networks that learn martingales.- Permutation recovery of spikes in noisy high-dimensional tensor estimation.- A User s Guide to KSig: GPU-Accelerated Computation of the Signature Kernel.