On Rough Paths and Signatures of Paths
Szymon Wirkus, Master student @ Department of Mathematics, University of Bergen
Abstract:
The aim of this seminar talk is to explore the paths in R^d of bounded p-variation and their role in extending integrals beyond the classical smooth setting. After presenting some standard results and discussing their further properties, we will be shifting our focus how different values of p determine the appropriate integration theory. In particular, we will examine the classical Riemann Stieltjes integral and Young integral, and the need of developing new theory whenever p>=2.
As a key concept, we introduce the concept of path signatures, viewed as a collection of iterated integrals. This leads to the framework developed by Terry Lyons, and further investigated by Peter K. Friz and Nicolas B. Victoir, as a sufficient tool to discuss the idea of integrating and solving differential equations driven by highly irregular signals. Lastly, we end by discussing how those concepts may be pushed to an infinite dimensional setting in a general Hilbert space.