The STOCHASTICA SNIP seminar will be held online every two weeks starting on January 14. Talks of 30-45 minutes will be followed by 15-30 minutes of open discussion. Initially, seminars will be focused on introducing new problems and applications, to stimulate further discussions and ideas within individual working groups.  

The fourth seminar was given by Prof Sonja Cox, University of Amsterdam, and chaired by Prof John Appleby (Leader, Working Group 1).

Universal approximation theorems for functions of infinite-dimensional paths

We establish universal approximation theorems for infinite-dimensional geometric rough paths, i.e., we show that continuous functions on the space of infinite-dimensional weakly geometric Hölder continuous rough paths can be approximated by functions that are linear in the signature of the path. The underlying topology determining continuity and compactness can be either the norm topology or a weak-* topology. Whereas considerably more effort is required to obtain the universal approximation theorem with respect to the weak-* topology, this setting ensures uniform approximation on norm-bounded sets. The motivation for establishing universal approximation theorems lies in the desire to approximate quantities derived from the solution of a stochastic partial differential equation. More specifically, our universal approximation theorems form the foundations of a novel approach to e.g. pricing of forward rates within the Heath-Jarrow-Morton-Musiela framework. This concerns joint work with Asma Khedher and Thijs Maessen.