A PDE approach for solving the characteristic function of the generalised signature process

The signature of a path, as a fundamental object in Rough path theory, serves as a generating function for non-commutative monomials on path space. It transforms the path into a grouplike element in the tensor algebra space, summarising the path faithfully up to a generalised form of re-parameterisation (a negligible equivalence class in this context). Our paper concerns stochastic processes and studies the characteristic function of the path signature of the stochastic process. 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 paper, we focus on a time-homogeneous It\^o diffusion process, and adopt a PDE approach to derive the characteristic function of its signature defined at any fixed time horizon. A key ingredient of our approach is the introduction of the generalised-signature process. This lifting enables us to establish the Feynman-Kac-type theorem for the characteristic function of the generalised-signature process by following the martingale approach. Moreover, as an application of our results, we present a novel derivation of the joint characteristic function of Brownian motion coupled with the L\'evy area, leveraging the structure theorem of anti-symmetric matrices.