Dependence properties and comparison results for Lévy processes |
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Authors: | Nicole Bäuerle Anja Blatter Alfred Müller |
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Institution: | 1.Institut für Stochastik,Universit?t Karlsruhe (TH),Karlsruhe,Germany;2.Department of Actuarial Mathematics and Statistics,Heriot-Watt University,Edinburgh,UK |
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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.
Anja Blatter was supported by the Deutsche Forschungsgemeinschaft (DFG). |
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Keywords: | Lévy processes Dependence concepts Lévy copula Dependence ordering Archimedean copula Ruin times Option pricing |
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