Universal approximation with signatures of non-geometric rough paths

We establish a universal approximation theorem for signatures of rough paths that are not necessarily weakly geometric. By extending the path with time and its rough path bracket terms, we prove that linear functionals of the signature of the resulting rough paths approximate continuous functionals on rough path spaces uniformly on compact sets. Moreover, we construct the signature of a path extended by its pathwise quadratic variation terms based on general pathwise stochastic integration \`a la F\"ollmer, in particular, allowing for pathwise It\^o, Stratonovich, and backward It\^o integration. In a probabilistic setting, we obtain a universal approximation result for linear functionals of the signature of continuous semimartingales extended by the quadratic variation terms, defined via stochastic It\^o integration. Numerical examples illustrate the use of signatures when the path is extended by time and quadratic variation in the context of model calibration and option pricing in mathematical finance.