A note on the family of extremality stochastic orders

- Each extremality stochastic order is generated by a partial order and its upper quadrant sets.
- The extremality stochastic orders are order-isomorphic, in particular order-isomorphic to the upper orthant order.
- The maximal generator of each extremality order is obtained.
- A new family of stochastic orders is defined by means of the partial orders which determine the extremality orders.

The family of extremality stochastic orders was introduced in Laniado et al. (2012) (Portfolio selection through an extremality stochastic order. Insurance: Mathematics and Economics 51, 1-9), as an extension of the upper and lower orthant orders, having important applications in the research of optimal allocations of wealth among risks in single period portfolio problems. In this paper we analyze some properties of such a family of stochastic orders, namely we prove that any extremality stochastic order is generated by a partial order on the Euclidean space and the class of upper quadrant sets of the partial order, showing that all the extremality orders are order-isomorphic. The above analysis will lead to the determination of the maximal generator of each extremality order by means of the maximal generator of the upper orthant order. Moreover we introduce a new family of stochastic orders which arises from the previous construction.