  In this paper, we give three equivalent properties of the class of
multivariate simple cubic natural exponential families (NEF's). The first
property says that the cumulant function of any basis of the family is a
solution of some Monge-Amp\'{e}re equation, the second is that the variance
function satisfies a differential equation, and the third is characterized by
the equality between two families of prior distributions related to the NEF.
These properties represent the extensions to this class of the properties
stated in $\cite{Casalis(1996)}$ and satisfied by the Wishart and the simple
quadratic NEF's. We also show that in the real case, each of these properties
provides a new characterization of the Letac-Mora class of real cubic NEF's.
