Weak Subordination of Multivariate L\'evy Processes and Variance Generalised Gamma Convolutions

TL;DR

This paper introduces a new weak subordination method for multivariate Lévy processes, extending traditional subordination techniques and broadening the class of processes with complex dependence structures, with applications in financial modeling.

Contribution

It proposes a novel weak subordination operation for Lévy processes that allows dependence among components, extending existing subordination methods and applications in variance-gamma processes.

Findings

Wider dependence structures in variance-$\alpha$-gamma processes.

Extension of variance generalised gamma convolution class.

Application of weak subordination to Brownian motion with Thorin subordinators.

Abstract

Subordinating a multivariate L\'evy process, the subordinate, with a univariate subordinator gives rise to a pathwise construction of a new L\'evy process, provided the subordinator and the subordinate are independent processes. The variance-gamma model in finance was generated accordingly from a Brownian motion and a gamma process. Alternatively, multivariate subordination can be used to create L\'evy processes, but this requires the subordinate to have independent components. In this paper, we show that there exists another operation acting on pairs $(T,X)$ of L\'evy processes which creates a L\'evy process $X\odot T$. Here, $T$ is a subordinator, but $X$ is an arbitrary L\'evy process with possibly dependent components. We show that this method is an extension of both univariate and multivariate subordination and provide two applications. We illustrate our methods giving a weak formulation of the variance-$\alpha$-gamma process that exhibits a wider range of dependence than using traditional subordination. Also, the variance generalised gamma convolution class of L\'evy processes formed by subordinating Brownian motion with Thorin subordinators is further extended using weak subordination.