Open Access
February 2019 Weak subordination of multivariate Lévy processes and variance generalised gamma convolutions
Boris Buchmann, Kevin W. Lu, Dilip B. Madan
Bernoulli 25(1): 742-770 (February 2019). DOI: 10.3150/17-BEJ1004


Subordinating a multivariate Lévy process, the subordinate, with a univariate subordinator gives rise to a pathwise construction of a new Lévy 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 Lévy 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 Lévy processes which creates a Lévy process $X\odot T$. Here, $T$ is a subordinator, but $X$ is an arbitrary Lévy 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-$\boldsymbol{\alpha}$-gamma process that exhibits a wider range of dependence than using traditional subordination. Also, the variance generalised gamma convolution class of Lévy processes formed by subordinating Brownian motion with Thorin subordinators is further extended using weak subordination.


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Boris Buchmann. Kevin W. Lu. Dilip B. Madan. "Weak subordination of multivariate Lévy processes and variance generalised gamma convolutions." Bernoulli 25 (1) 742 - 770, February 2019.


Received: 1 June 2017; Revised: 1 October 2017; Published: February 2019
First available in Project Euclid: 12 December 2018

zbMATH: 07007223
MathSciNet: MR3892335
Digital Object Identifier: 10.3150/17-BEJ1004

Keywords: Brownian motion , gamma process , Generalised gamma convolutions , Lévy process , marked point process , Subordination , Thorin measure , variance gamma , variance-alpha-gamma

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 1 • February 2019
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