Some Results on the Additivity and Multiplication Order Preserving Properties of Stochastic Orders

In this paper, we consider the problem of order preservation under addition and multiplication operators over the vector space of univariate real-valued random variables. Consistent with the case of usual order over the real numbers-as constant random variables, we prove that the usual stochastic order is both additive and multiplicative. We additionally discuss the situation for nine extra univariate stochastic orders presenting counterexamples in some cases. The problem remains unsolved for other univariate stochastic orders.