Tao, Jiajie; (2025) Study of the characteristic function of stochastic processes: theory and applications. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Characterizing measures on the path space has significant theoretical implications in stochastic analysis and practical applications in machine learning. Rough Path Theory offers two faithful feature representations of a stochastic process: (1) the characteristic function of the signature and (2) the path characteristic function (PCF). These two concepts generalize the classical characteristic functions of random variables to stochastic processes from different yet interconnected perspectives. In this thesis, we aim to tackle three key research questions regarding these two features. Firstly, we are concerned with the characteristic function of the signature of the time-homogeneous Itô diffusion process and establish the Feynman-Kac type theorem, which enables solving it via the PDE approach. We further extend this approach for the underlying process, which is of signature SDE type and goes far beyond diffusion processes. As a byproduct, this approach results in a novel proof to recover the classical results on the characteristic function of the Brownian motion with its Lévy area at a fixed time. Secondly, we investigate the extended weak topology (EWT) of stochastic processes, which not only characterizes the law on the path space but also captures its filtration. We propose a novel distance, the so-called High-Rank Path Characteristic Function Distance (HRPCFD), that metrizes EWT, and we prove its favourable analytic properties. Additionally, we design efficient algorithms to allow their successful application to both hypothesis testing on stochastic processes and synthetic time series generation, with significantly superior performance improvements. Lastly, we consider the task of developing high-fidelity generative models for generating the Lévy area of Brownian motion to accelerate the numerical solver of SDEs. We introduce the mathematically principled Lévy-GAN model, with one of its main innovations being a discriminator inspired by the PCF. A key advantage of this model is that it does not require training data, offering a significant step forward in accuracy and efficiency.

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