Bernoulli

  • Bernoulli
  • Volume 25, Number 1 (2019), 742-770.

Weak subordination of multivariate Lévy processes and variance generalised gamma convolutions

Boris Buchmann, Kevin W. Lu, and Dilip B. Madan

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Abstract

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.

Article information

Source
Bernoulli, Volume 25, Number 1 (2019), 742-770.

Dates
Received: June 2017
Revised: October 2017
First available in Project Euclid: 12 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1544605262

Digital Object Identifier
doi:10.3150/17-BEJ1004

Mathematical Reviews number (MathSciNet)
MR3892335

Zentralblatt MATH identifier
07007223

Keywords
Brownian motion gamma process generalised gamma convolutions Lévy process marked point process subordination Thorin measure variance gamma variance-alpha-gamma

Citation

Buchmann, Boris; Lu, Kevin W.; Madan, Dilip B. Weak subordination of multivariate Lévy processes and variance generalised gamma convolutions. Bernoulli 25 (2019), no. 1, 742--770. doi:10.3150/17-BEJ1004. https://projecteuclid.org/euclid.bj/1544605262


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