Directional Stochastic Orders with an Application to Financial Mathematics

Abstract

Relevant integral stochastic orders share a common mathematical model, they are defined by generators which are made up of increasing functions on appropriate directions. Motivated by the aim to provide a unified study of those orders, we introduce a new class of integral stochastic orders whose generators are composed of functions that are increasing on the directions of a finite number of vectors. These orders will be called directional stochastic orders. Such stochastic orders are studied in depth. In that analysis, the conical combinations of vectors in those finite subsets play a relevant role. It is proved that directional stochastic orders are generated by non-stochastic pre-orders and the class of their preserving mappings. Geometrical characterizations of directional stochastic orders are developed. Those characterizations depend on the existence of non-trivial subspaces contained in the set of conical combinations. An application of directional stochastic orders to the field of financial mathematics is developed, namely, to the comparison of investments with random cash flows.

Keywords

• convex cone
• investment with random cash flows
• non-trivial subspace
• orthogonal projection
• stochastic order