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January 2007 Comparison of semimartingales and Lévy processes
Jan Bergenthum, Ludger Rüschendorf
Ann. Probab. 35(1): 228-254 (January 2007). DOI: 10.1214/009117906000000386


In this paper, we derive comparison results for terminal values of d-dimensional special semimartingales and also for finite-dimensional distributions of multivariate Lévy processes. The comparison is with respect to nondecreasing, (increasing) convex, (increasing) directionally convex and (increasing) supermodular functions. We use three different approaches. In the first approach, we give sufficient conditions on the local predictable characteristics that imply ordering of terminal values of semimartingales. This generalizes some recent convex comparison results of exponential models in [Math. Finance 8 (1998) 93–126, Finance Stoch. 4 (2000) 209–222, Proc. Steklov Inst. Math. 237 (2002) 73–113, Finance Stoch. 10 (2006) 222–249]. In the second part, we give comparison results for finite-dimensional distributions of Lévy processes with infinite Lévy measure. In the first step, we derive a comparison result for Markov processes based on a monotone separating transition kernel. By a coupling argument, we get an application to the comparison of compound Poisson processes. These comparisons are then extended by an approximation argument to the ordering of Lévy processes with infinite Lévy measure. The third approach is based on mixing representations which are known for several relevant distribution classes. We discuss this approach in detail for the comparison of generalized hyperbolic distributions and for normal inverse Gaussian processes.


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Jan Bergenthum. Ludger Rüschendorf. "Comparison of semimartingales and Lévy processes." Ann. Probab. 35 (1) 228 - 254, January 2007.


Published: January 2007
First available in Project Euclid: 19 March 2007

zbMATH: 1178.60035
MathSciNet: MR2303949
Digital Object Identifier: 10.1214/009117906000000386

Primary: 60E15
Secondary: 60G44 , 60G51

Keywords: Convex ordering , jump diffusion process , Lévy processes , propagation of convexity

Rights: Copyright © 2007 Institute of Mathematical Statistics


Vol.35 • No. 1 • January 2007
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