Universal approximation with signatures of non-geometric rough paths
This work addresses a foundational challenge in rough path theory for researchers in stochastic analysis and mathematical finance, offering a novel extension beyond weakly geometric paths.
The paper tackles the problem of approximating continuous functionals on rough path spaces by proving a universal approximation theorem for signatures of non-geometric rough paths, using extensions with time and bracket terms, and demonstrates applications in mathematical finance with numerical examples for model calibration and option pricing.
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 à la Föllmer, in particular, allowing for pathwise Itô, Stratonovich, and backward Itô 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 Itô 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.