Based on the idea of constructing a time-changed process, subordination is the operation that evaluates a Lévy process at a subordinator. This produces a Lévy process when the subordinate has independent components or the subordinator has indistinguishable components, otherwise it may not. Weak subordination will be introduced, extending this idea in a way that always produces a Lévy process and matches traditional subordination in law in the previous cases. We show that variance generalised gamma convolutions, which are processes formed by the weak subordination of Brownian motion with a Thorin subordinator are self-decomposable if the Brownian motion is drift-less, and the converse holds under moment conditions on the Thorin measure. This class includes a wide range of processes used in mathematical finance to model dependence across multiple prices. One example of the latter is the weak variance-alpha-gamma process, and we study methods for estimating its parameters despite the density function not being explicitly known. In our simulations, we find that maximum likelihood produces a better fit when a Fourier invertibility condition holds, while digital moment estimation produces a better fit when it doesn't. The weak variance-alpha-gamma process exhibits a wider range of dependence and produces a significantly better fit than its traditional subordination counterpart and is fitted to stock price data.
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