Regulating Functions on Partially Ordered Sets

Abstract

We study the so-called Skorokhod reflection problem (SRP) posed for real-valued functions defined on a partially ordered set (poset), when there are two boundaries, considered also to be functions of the poset. The problem is to constrain the function between the boundaries by adding and subtracting nonnegative nondecreasing (NN) functions in the most efficient way. We show existence and uniqueness of its solution by using only order theoretic arguments. The solution is also shown to obey a fixed point equation. When the underlying poset is a σ-algebra of subsets of a set, our results yield a generalization of the classical Jordan–Hahn decomposition of a signed measure. We also study the problem on a poset that has the structure of a tree, where we identify additional structural properties of the solution, and on discrete posets, where we show that the fixed point equation uniquely characterizes the solution. Further interesting posets we consider are the poset of real n-vectors ordered by majorization, and the poset of n × n positive semidefinite real matrices ordered by pointwise ordering of the associated quadratic forms. We say a function on a poset is of bounded variation if it can be written as the difference of two NN functions. The solution to the SRP when the upper and lower boundaries are the identically zero function corresponds to the most efficient or minimal such representation of a function of bounded variation. Minimal representations for several important functions of bounded variation on several of the posets mentioned above are determined in this paper.