The key purpose of this paper is to present Fourier method to model the stochastic time-change in this context of time-subordinated Brownian motion models. We review Gaussian Variance-Mean mixtures and time-subordinated models with a key example of the Gamma process. A non-parametric characteristic function decomposition of subordinated Brownian motion is presented. This allows one to characterise and study the stochastic time-change directly from the full process. Finally we provide an example empirical decomposition of S$\&$P log-returns. We explore the Variance Gamma process as a key example throughout.

翻译：本文的核心目的是提出一种傅里叶方法，用于在时从属布朗运动模型的背景下对随机时间变化进行建模。我们回顾了高斯方差-均值混合模型以及时从属模型，并以Gamma过程作为关键示例。本文提出了一种从属布朗运动的非参数特征函数分解方法。这使得人们能够直接从完整过程出发来刻画和研究随机时间变化。最后，我们以标准普尔指数对数收益为例，给出了一个经验分解的示例。全文以方差Gamma过程作为贯穿始终的核心示例进行探讨。