Skip to main content
SpringerLink
Log in

Series Representations for Multivariate Time-Changed Lévy Models

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

In this paper, we analyze a Lévy model based on two popular concepts - subordination and Lévy copulas. More precisely, we consider a two-dimensional Lévy process such that each component is a time-changed (subordinated) Brownian motion and the dependence between subordinators is described via some Lévy copula. The main result of this paper is the series representation for our model, which can be efficiently used for simulation purposes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Ané T, Geman H (2000) Order flow, transaction clock, and normality of asset returns. J Financ 55(5):2259–2284

  • Avanzi B, Cassar L, Wong B (2011) Modelling dependence in insurance claims processes with Lévy copulas. ASTIN Bulletin 41:575–609

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen O, Lindner A (2004) Some aspects of Lévy copulas. SFB 386, Discussion paper 388

  • Barndorff-Nielsen O, Pedersen J, Sato K (2001) Multivariate subordination, self-decomposability and stability. Advances in Applied Probability 33:160–187

    Article  MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen OE, Shiryaev AN (2010) Change of Time and Change of Measure. World Scientific

  • Barndorff-Nielsen O, Kent J, Sorensen M (1982) Normal variance-mean mixtures and z distributions. International statistical review 50:145–159

    Article  MathSciNet  MATH  Google Scholar 

  • Belomestny D, Panov V (2013) Estimation of the activity of jumps in time-changed Lévy models. Electron. J. Statist 7:2970–3003. doi:10.1214/13-EJS870

    Article  MATH  Google Scholar 

  • Bertoin J (1998) Lévy processes. Cambridge University Press

  • Bücher A, Vetter M (2013) Nonparametric inference on Lévy measures and copulas. The Annals of Statistics 41:1485–1515

    Article  MathSciNet  MATH  Google Scholar 

  • Carr P, Geman H, Madan D, Yor M (2003) Stochastic volatility for Lévy processes. Mathematical Finance 13:345–382

    Article  MathSciNet  MATH  Google Scholar 

  • Cherny AS, Shiryaev AN (2002) Change of time and measure for Lévy processes. Lectures for the summer school “From Lévy processes to semimartingales - recent theoretical developments and applications to finance”

  • Cherubini U, Lunga GD, Mulinacci S, Rossi P (2010) Fourier transform methods in finance. Wiley

  • Cherubini U, Lunga E, Vecchiato W (2004) Copula methods in finance. Wiley

  • Clark PK (1973) A subordinated stochastic process model with fixed variance for speculative prices. Econometrica 41:135–156

    Article  MathSciNet  MATH  Google Scholar 

  • Cohen S, Rosinski J (2007) Gaussian approximation of multivariate Lévy processes with applications to simulation of tempered stable processes. Bernoulli 13:195–210. doi:10.3150/07-BEJ6011 2307403 (2008a:60120)

    Article  MathSciNet  MATH  Google Scholar 

  • Cont R, Tankov P (2004) Financial modelling with jump process. Chapman & Hall, CRC Press, UK

    MATH  Google Scholar 

  • Eberlein E, Madan D (2010) On correlating Lévy processes. The Journal of Risk 13:3–16

    Article  Google Scholar 

  • Esmaeli H, Klüppelberg C (2010) Parameter estimation of a bivariate compound Poisson process. Insurance: mathematics and economics 47:224–233

    MathSciNet  MATH  Google Scholar 

  • Esmaeli H, Klüppelberg C (2011) Parametric estimation of a bivariate stable Lévy process. Journal of Multivariate Analysis 102:918–930

    Article  MathSciNet  MATH  Google Scholar 

  • Griffin JE, Walker SG (2011) Posterior simulation of normalized random measure mixtures. 241–259 20. Supplementary material available online. doi:10.1198/jcgs.2010.08176 2816547 (2012c:62078)

  • Grothe O, Hofert M (2015) Construction and sampling of Archimedean and nested Archimedean Lévy copulas. J. Multivariate Anal 138:182–198. doi:10.1016/j.jmva.2014.12.004 3348841

    Article  MathSciNet  MATH  Google Scholar 

  • Grothe O, Nicklas S (2013) Vine constructions of Lévy copulas. J. Multivariate Anal 119:1–15. doi:10.1016/j.jmva.2013.04.002 3061411

    Article  MathSciNet  MATH  Google Scholar 

  • Hilber RNSCN, Winter C (2009) Numerical methods for Lévy processes. Finance Stoch 13:471–500. doi:10.1007/s00780-009-0100-5 2519841 (2010i:60235)

    Article  MathSciNet  MATH  Google Scholar 

  • Ishwaran H, Zarepour M (2009) Series representations for multivariate generalized gamma processes via a scale invariance principle. Statist. Sinica 19:1665–1682. 2589203 (2010m:60169)

    MathSciNet  MATH  Google Scholar 

  • Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall

  • Kalli M, Griffin JE, Walker SG (2011) Slice sampling mixture models. Stat. Comput 21:93–105. doi:10.1007/s11222-009-9150-y 2746606

    Article  MathSciNet  MATH  Google Scholar 

  • Kallsen J, Tankov P (2006) Characterization of dependence of multidimensional Lévy processes using Lévy copulas. Journal of Multivariate Analysis 97:1551–1572

    Article  MathSciNet  MATH  Google Scholar 

  • Kelker D (1971) Infinite divisibility and variance mixtures of the normal distribution. The Annals of Mathematical Statistics 42:802–808

    Article  MathSciNet  MATH  Google Scholar 

  • Kolossiatis M, Griffin JE, Steel MFJ (2013) On Bayesian nonparametric modelling of two correlated distributions. Stat. Comput 23:1–15. doi:10.1007/s11222-011-9283-7 3018346

    Article  MathSciNet  MATH  Google Scholar 

  • Leisen F, Lijoi A (2011) Vectors of two-parameter Poisson-Dirichlet processes. J. Multivariate Anal 102:482–495. doi:10.1016/j.jmva.2010.10.008 2755010 (2011j:62074)

    Article  MathSciNet  MATH  Google Scholar 

  • Leisen F, Lijoi A, Spanó D (2013) A vector of Dirichlet processes. Electron. J. Stat 7:62–90. doi:10.1214/12-EJS764 3020414

    Article  MathSciNet  MATH  Google Scholar 

  • Luciano E, Semeraro P (2010) Multivariate time changes for Lévy asset models: characterization and calibration. J. Comput. Appl. Math 233:1937–1953. doi:10.1016/j.cam.2009.08.119 2564029 (2011b:91280)

    Article  MathSciNet  MATH  Google Scholar 

  • Monroe I (1978) Processes that can be embedded in Brownian motion. The Annals of Probability 6:42–56

    Article  MathSciNet  MATH  Google Scholar 

  • Nelsen R (2006) An introduction to copulas, 2nd Edn. Springer

  • Rosinski J (2001) Series representations of Lévy processes from the perspective of point processes. In: Barndorff-Nielsen O, Mikosch T, Resnick S (eds) Lévy processes: theory and applications. Springer Science+Business Media

  • Sato K (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press

  • Schoutens W (2003) Lévy processes in finance. Wiley

  • Semeraro P (2008) A multivariate variance gamma model for financial applications. International journal of theoretical and applied finance 11:1–18

    Article  MathSciNet  MATH  Google Scholar 

  • Tankov P (2003) Dependence structure of spectrally positive multidimensional Lévy processes. Preprint

  • Tankov P (2004) Lévy processes in finance: inverse problems and dependence modelling, PhD thesis. Ecole Polytechnique, Palaiseau

  • Panov V, Sirotkin I (2015) Series representations for bivariate time-changed Lévy models. arXiv:1503.02214

  • Veraart A, Winkel M (2010) Time change. In: Cont R (ed) Encyclpedia of quantitative finance. Wiley

  • Zhu W, Leisen F (2015) A multivariate extension of a vector of two-parameter Poisson-Dirichlet processes. J. Nonparametr. Stat 27:89–105. doi:10.1080/10485252.2014.966103 3304361

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Research University Higher School of Economics, Shabolovka 31, building G, 115162, Moscow, Russia

    Vladimir Panov

Authors
  1. Vladimir Panov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir Panov.

Additional information

This study (research grant No 14-05-0007) was supported by the National Research University-Higher School of Economics’ Academic Fund Program in 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Panov, V. Series Representations for Multivariate Time-Changed Lévy Models. Methodol Comput Appl Probab 19, 97–119 (2017). https://doi.org/10.1007/s11009-015-9461-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-015-9461-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation