06.06.2023, 15 Uhr
Phenomena in High Dimensions
Pierre Youssef (New York University, Abu Dhabi)
Ilya Chevyrev, TU
An introduction to rough paths
Abstract: The theory of rough paths has been developed over the past two decades in order to give a pathwise approach to classically ill-posed controlled differential equations. In contrast to Ito calculus, where solutions to differential equations are constructed in an intrinsically probabilistic manner, rough paths theory aims to separate the analytic and probabilistic steps and define solutions as continuous functions of the driving noise. Over the course of the talks, we aim to provide an introduction to this theory.
We will first review controlled differential equations, and in particular demonstrate how the classical method of Young integration fails to apply for sufficiently rough input signals which arise naturally in stochastic analysis. We then plan to introduce the notion of a rough path which allows us to construct pathwise solutions to controlled differential equations with rough inputs. We will then show the central result in rough paths theory, the universal limit theorem, which demonstrates that the so-constructed solutions are in fact continuous functions of the input. We will conclude with a simple application to stochastic differential equations by demonstrating strong and weak convergence results.