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

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.