Dependence properties and comparison results for Lévy processes

Nicole Bäuerle, Anja Blatter, Alfred Müller

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

In this paper we investigate dependence properties and comparison results for multidimensional Lévy processes. In particular we address the questions, whether or not dependence properties and orderings of the copulas of the distributions of a Lévy process can be characterized by corresponding properties of the Lévy copula, a concept which has been introduced recently in Cont and Tankov (Financial modelling with jump processes. Chapman & Hall/CRC, Boca Raton, 2004) and Kallsen and Tankov (J Multivariate Anal 97:1551-1572, 2006). It turns out that association, positive orthant dependence and positive supermodular dependence of Lévy processes can be characterized in terms of the Lévy measure as well as in terms of the Lévy copula. As far as comparisons of Lévy processes are concerned we consider the supermodular and the concordance order and characterize them by orders of the Lévy measures and by orders of the Lévy copulas, respectively. An example is given that the Lévy copula does not determine dependence concepts like multivariate total positivity of order 2 or conditionally increasing in sequence. Besides these general results we specialize our findings for subfamilies of Lévy processes. The last section contains some applications in finance and insurance like comparison statements for ruin times, ruin probabilities and option prices which extends the current literature. © 2007 Springer-Verlag.

Original languageEnglish
Pages (from-to)161-186
Number of pages26
JournalMathematical Methods of Operations Research
Volume67
Issue number1
DOIs
Publication statusPublished - Feb 2008

Keywords

  • Archimedean copula
  • Dependence concepts
  • Dependence ordering
  • Lévy copula
  • Lévy processes
  • Option pricing
  • Ruin times

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