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.
Highlights
This manuscript is focused on the theory of stochastic orders
Some integral stochastic orders have a common condition in relation to the functions of their generators, these are composed of functions which are increasing on appropriate directions
In order to develop a general analysis of integral stochastic orders satisfying the above condition, we introduce a class of integral stochastic orders with generators which are made up of increasing functions on the directions given by finite subsets of vectors in Rn
Summary
This manuscript is focused on the theory of stochastic orders. A lot of effort has been made on this topic during the last decades due to its importance from the theoretical and applied points of view. Some integral stochastic orders have a common condition in relation to the functions of their generators, these are composed of functions which are increasing on appropriate directions. (ii) the time value of money stochastic order, a generator of that stochastic order is given by the functions which are increasing on the directions of the vectors in {e1, e2, . The class considered in the manuscript permits to introduce new orders (as we develop, to approach a problem in financial mathematics), the properties of those orders being immediate to state by the theoretical study of the paper. Our research plan will be focused on the analysis of the relations between two directional stochastic orders given by two different finite subsets of vectors in Rn (Section 3), the development of geometrical characterizations of directional orders (Section 4), and the study of relevant properties of those orders (Section 5). A new directional stochastic order for the comparison of investments with random cash flows is introduced in Section 6, as an application of the results in the present manuscript
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