Conditions for the Multivariate Stochastic Order Under Dependence Uncertainty

In the univariate case checking the usual stochastic order for two random variables is equivalent to comparing point-wise their distribution functions.Therefore, the task is easy.The multivariate case this is much more complicated to handle, because it requires the comparison of the probability of a huge class of sets (all the upper sets). In this paper we consider conditions for multivariate stochastic order for the case that we only know the marginal distributions and have no or only partial knowledge of the dependence structure.