Universal approximation with signatures of non-geometric rough paths

Recently, data-driven methods based on path signatures have gained prominence in mathematical finance. They rely on universal approximation theorems stating that continuous functionals on path space can be approximated uniformly on compact sets by linear functionals of the signature. In financial applications, this has led to the use of Stratonovich-signatures, although Itô integration is often the natural modeling framework.

In this talk, we establish a universality result for signatures of non-geometric rough paths. By augmenting the path with its rough path bracket, we obtain a quasi-shuffle structure that provides the algebraic basis for universality. For continuous semimartingales, this yields a universal approximation property for Itô signatures.

This talk is based on joint work with A. P. Kwossek and D. J. Prömel.