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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Applebaum D (2004). Lévy processes and stochastic calculus. Cambridge University Press, Cambridge
Asmussen S, Frey A, Rolski T, Schmidt V (1995) Does Markov-modulation increase the risk? ASTIN Bull 49–66
Bäuerle N (1997). Inequalities for stochastic models via supermodular orderings. Comm Stat Stoch Models 13: 181–201
Bäuerle N (2002). Risk management in credit risk portfolios with correlated assets. Insur Math Econ 30: 187–198
Bäuerle N and Rolski T (1998). A monotonicity result for the workload in Markov-modulated queues. J Appl Probab 35: 741–747
Bergenthum J and Rüschendorf L (2007). Comparison of Semimartingales and Lévy processes. Ann Probab 35: 228–254
Bregman Y and Klüppelberg C (2005). Ruin estimation in multivariate models with Clayton dependence structure. Scand Actuar J 2005: 462–480
Christofides TC and Vaggelatou E (2004). A connection between supermodular ordering and positive/negative association. J Multivariate Anal 88: 138–151
Cont R and Tankov P (2004). Financial modelling with jump processes. Chapman & Hall/CRC, Boca Raton
Denuit M and Müller A (2002). Smooth generators of integral stochastic orders. Ann Appl Probab 12: 1174–1184
Denuit M, Dhaene J, Goovaerts M and Kaas R (2005). Actuarial theory for dependent risks: measures, orders and models. Orders and models. Wiley, New York
Denuit M, Frostig E and Levikson B (2007). Supermodular comparison of time-to-ruin random vectors. Methodol Comput Appl Probab 9: 41–54
Ebrahimi N (2002). On the dependence structure of certain multi-dimensional Ito processes and corresponding hitting times. J Multivariate Anal 81: 128–137
Esary JD, Proschan F and Walkup DW (1967). Association of random variables, with applications. Ann Math Stat 38: 1466–1474
Houdré C (1998) Comparison and deviation from a representation formula. In: Stochastic processes and related topics. Trends Math. Birkhäuser Boston, Boston, pp 207–218
Houdré C, Pérez-Abreu V and Surgailis D (1998). Interpolation, correlation identities, and inequalities for infinitely divisible variables. J Fourier Anal Appl 4: 651–668
Joe H (1997) Multivariate models and dependence concepts. Monographs on statistics and applied probability, vol 73. Chapman & Hall, London
Juri A (2002) Supermodular order and Lundberg exponents. Scand Actuar J 17–36
Kallsen J and Tankov P (2006). Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J Multivariate Anal 97: 1551–1572
Karlin S and Rinott Y (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J Multivariate Anal 10: 467–498
Lehmann EL (1966). Some concepts of dependence. Ann Math Stat 37: 1137–1153
Liggett TM (2005). Interacting particle systems. Springer, Berlin
Müller A and Scarsini M (2000). Some remarks on the supermodular order. J Multivariate Anal 73: 107–119
Müller A and Stoyan D (2002). Comparison methods for stochastic models and risks. Wiley series in probability and statistics. Wiley, Chichester
Nelsen RB (2006). An introduction to copulas. Springer, New York
Protter P (1990). Stochastic integration and differential equations. Applications of mathematics (New York). Springer, Berlin
Resnick SI (1988). Association and multivariate extreme value distributions. Aust J Stat 30: 261–271
Samorodnitsky G (1995). Association of infinitely divisible random vectors. Stoch Process Appl 55: 45–55
Sato K-I (1999). Lévy processes and infinitely divisible distributions. Cambridge studies in advanced mathematics, vol 68. Cambridge University Press, Cambridge
Szekli R (1995). Stochastic ordering and dependence in applied probability. Lecture notes in statistics, vol 97. Springer, New York
Author information
Authors and Affiliations
Corresponding author
Additional information
Anja Blatter was supported by the Deutsche Forschungsgemeinschaft (DFG).
Rights and permissions
About this article
Cite this article
Bäuerle, N., Blatter, A. & Müller, A. Dependence properties and comparison results for Lévy processes. Math Meth Oper Res 67, 161–186 (2008). https://doi.org/10.1007/s00186-007-0185-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-007-0185-6
Keywords
- Lévy processes
- Dependence concepts
- Lévy copula
- Dependence ordering
- Archimedean copula
- Ruin times
- Option pricing