Abstract
Given a partial order ⪯ on a set X , one can consider the class of ⪯-preserving real functions on X characterized by x ⪯ y implies f ( x ) ≤ f ( y ) . Such a class of functions allows us the generation of a binary relation ⪯ g on the set of probabilities associated with X by means of P ⪯ g Q when ∫ f d P ≤ ∫ f d Q for all ⪯-preserving functions f. In this paper we characterize when for an integral stochastic order ⪯ ˜ on the set of probabilities associated with X , there exists a partial order ⪯ on X such that the relation ⪯ g generated by the class of ⪯-preserving functions is equal to ⪯ ˜ . The above characterization is related to the maximal generator of ⪯ ˜ , a result which can be applied for the search of maximal generators of stochastic orders generated by posets. MSC:06A06, 60E15.
Highlights
Stochastic orders play a key role in many areas
A stochastic order is defined as a partial order relation on a set of probabilities associated with a certain measurable space, in some contexts the antisymmetric condition is not considered
We briefly describe the concept of a maximal generator of an integral stochastic order
Summary
Stochastic orders play a key role in many areas. They have been applied successfully in such fields as reliability theory, economics, decision theory, queueing systems, scheduling problems, medicine, genetics, etc. A stochastic order on P is said to be integral if there exists a class R of measurable mappings from X to R satisfying that P Q if and only if f dP ≤ f dQ Given an integral stochastic order, we will denote by the partial order on X determined by , that is, x y when f (x) ≤ f (y) for all mappings of a generator of .
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