Abstract: The signature of a path, as a fundamental object in Rough path theory, serves as non-communicative monomials on the path space. It transforms the path into a grouplike element in the tensor algebra space, summarizing the path faithfully up to a negligible equivalence class. Our work concerns the characteristic function of the signature of stochastic processes. In contrast to the expected signature, it determines the law on the random signatures without any regularity condition. The computation of the characteristic function of the random signature offers potential applications in stochastic analysis and machine learning, where the expected signature plays an important role. In this talk, we focus on a time-homogeneous Ito diffusion process, adopting a PDE approach to derive the characteristic function of its signature defined at any fixed time horizon. By using this approach, we can recover the results of the characteristic function of the signature SDEs, which go beyond the diffusion setting. As an application of our method, we present a novel derivation of the joint characteristic function of Brownian motion coupled with the Levy area, leveraging the structure theorem of anti-symmetric matrices. Lastly, we conclude the presentation with its application to generative modelling for high-fidelity Levy area simulation.

This talk is based on two preprints:

[1] Lyons, T., Ni, H. and Tao, J., 2024. A PDE approach for solving the characteristic function of the generalised signature process. arXiv preprint arXiv:2401.02393.

[2] Jelinčič, A., Tao, J., Turner, W.F., Cass, T., Foster, J. and Ni, H., 2023. Generative Modelling of Lévy Area for High Order SDE Simulation. arXiv preprint arXiv:2308.02452.