On Time-subordinated Brownian Motion Processes for Financial Markets

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.