Discrete Higher Order Convex Functions and their Applications

Abstract

In this paper we present an overview about the recently developed theory of discrete moment problems, i.e., moment problems where the supports of the random variables involved are discrete. We look for the minimum or maximum of a linear functional acting on an unknown probability distribution subject to a finite number of moment constraints. Using linear programming methodology, we present structural theorems, in both the univariate and multivariate cases, for the dual feasible bases and show how the relevant problems can be solved by suitable adaptations of the dual method. The condition on the objective function is a kind of higher order convexity, expressed in terms of divided differences. A variant of the above, the discrete binomial moment problem, as well as generalization for discrete variable Chebyshev systems are also discussed. Finally, we present novel applications to valuations of financial instruments.