  In this paper, we derive comparison results for terminal values of
$d$-dimensional special semimartingales and also for finite-dimensional
distributions of multivariate L\'{e}vy 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 L\'{e}vy processes with infinite L\'{e}vy
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
L\'{e}vy processes with infinite L\'{e}vy 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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