Among the different methods proposed to lift path-dependent dynamics to infinite-dimensional Markovian frameworks, the use of signatures appears especially natural, as linear functionals of the signature can approximate any continuous path functional (with respect to suitable Hölder/variation topologies) arbitrarily well. While such universal approximation results at the level of the vector fields are well established, we go further and consider solutions of generic path-dependent controlled differential equations (CDEs). We then show that, under mild regularity assumptions, any such path-dependent system can be approximated by a suitable signature CDE. To this end we first establish well-posedness and stability of path-dependent systems using weighted space topologies for Hölder continuous paths. We then transfer these results to signature CDEs, deriving in particular well-posedness conditions and a dynamic universal approximation theorem when the vector fields are real-analytic functions of the signature.

This talk is based on joint work with Tomas Carrondo, Paul Hager, and Fabian Harang.