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Dependence properties and comparison results for Lévy processes. (English) Zbl 1141.60023

Summary: 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 by R. Cont and P. Tankov [Financial modelling with jump processes. Chapman & Hall/CRC, Boca Raton (2004 ; Zbl 1052.91043)] and J. Kallsen and P. Tankov [J. Multivariate Anal. 97, No. 7, 1551–1572 (2006; Zbl 1099.62048)]. 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.

MSC:

60G51 Processes with independent increments; Lévy processes
60E15 Inequalities; stochastic orderings
62H05 Characterization and structure theory for multivariate probability distributions; copulas
91B28 Finance etc. (MSC2000)
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