A PDE approach for solving the characteristic function of the generalised signature process
arxiv(2024)
摘要
The signature of a path, as a fundamental object in Rough path theory, serves
as a generating function for non-communicative 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 Itô 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 Lévy area,
leveraging the structure theorem of anti-symmetric matrices.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要