Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T20:26:28.505Z Has data issue: false hasContentIssue false
  • English
  • Français

Necessity of weak subordination for some strongly subordinated Lévy processes

Part of: Stochastic processes

Published online by Cambridge University Press:  22 November 2021

Boris Buchmann  and
Kevin W. Lu Open the ORCID record for Kevin W. Lu [Opens in a new window]
Boris Buchmann*
Affiliation:
Australian National University
Kevin W. Lu*
Affiliation:
Australian National University
*
*Postal address: Research School of Finance, Actuarial Studies & Statistics, Australian National University, ACT 0200, Australia. Email address: [email protected]
**Current address: Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider the strong subordination of a multivariate Lévy process with a multivariate subordinator. If the subordinate is a stack of independent Lévy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a Lévy process; otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a Lévy process even when strong subordination does not. Here we prove that strong and weak subordination are equal in law under the aforementioned condition. In addition, we prove that if strong subordination is a Lévy process then it is necessarily equal in law to weak subordination in two cases: firstly when the subordinator is deterministic, and secondly when it is pure-jump with finite activity.

Keywords

Lévy processsubordinatormultivariate subordinationweak subordinationPoisson point processPoisson random measure

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60G55: Point processes
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 868 - 879
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barndorff-Nielsen, O. E., Pedersen, J. and Sato, K. (2001). Multivariate subordination, self-decomposability and stability. Adv. Appl. Prob. 33, 160–187.10.1017/S0001867800010685CrossRefGoogle Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Buchmann, B., Kaehler, B., Maller, R. and Szimayer, A. (2017). Multivariate subordination using generalised Gamma convolutions with applications to Variance Gamma processes and option pricing. Stoch. Process. Appl. 127, 2208–2242.10.1016/j.spa.2016.10.008CrossRefGoogle Scholar
Buchmann, B., Lu, K. W. and Madan, D. B. (2019). Calibration for weak variance-alpha-gamma processes. Methodology Comput. Appl. Prob. 21, 1151–1164.10.1007/s11009-018-9655-yCrossRefGoogle Scholar
Buchmann, B., Lu, K. W. and Madan, D. B. (2019). Weak subordination of multivariate Lévy processes and variance generalised gamma convolutions. Bernoulli 25, 742–770.10.3150/17-BEJ1004CrossRefGoogle Scholar
Çinlar, E. (2011). Probability and Stochastics. Springer, New York.10.1007/978-0-387-87859-1CrossRefGoogle Scholar
Guillaume, F. (2013). The $\alpha$ VG model for multivariate asset pricing. Rev. Derivatives Res. 16, 25–52.10.1007/s11147-012-9080-2CrossRefGoogle Scholar
Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.Google Scholar
Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press.Google Scholar
Luciano, E. and Semeraro, P. (2010). Multivariate time changes for Lévy asset models: characterization and calibration. J. Comput. Appl. Math. 233, 1937–1953.10.1016/j.cam.2009.08.119CrossRefGoogle Scholar
Madan, D. B. (2018). Instantaneous portfolio theory. Quant. Finance 18, 1345–1364.10.1080/14697688.2017.1420210CrossRefGoogle Scholar
Madan, D. B. and Seneta, E. (1990). The variance gamma (v.g.) model for share market returns. J. Business 63, 511–524.10.1086/296519CrossRefGoogle Scholar
Michaelsen, M. and Szimayer, A. (2018). Marginal consistent dependence modeling using weak subordination for Brownian motions. Quant. Finance 18, 1909–1925.10.1080/14697688.2018.1439182CrossRefGoogle Scholar
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Semeraro, P. (2008). A multivariate variance gamma model for financial applications. Int. J. Theor. Appl. Finance 11, 1–18.10.1142/S0219024908004701CrossRefGoogle Scholar
Zolotarev, V. M. (1958). Distribution of the superposition of infinitely divisible processes. Theory Prob. Appl. 3, 185–188.10.1137/1103017CrossRefGoogle Scholar