Maxentropic Solutions to a Convex Interpolation Problem Motivated by Utility Theory

Here, we consider the following inverse problem: Determination of an increasing continuous function U(x) on an interval [a, b] from the knowledge of the integrals ∫ U(x)dFXi (x) = πi where the Xi are random variables taking values on [a, b] and πi are given numbers. This is a linear integral equation with discrete data, which can be transformed into a generalized moment problem when U(x) is supposed to have a positive derivative, and it becomes a classical interpolation problem if the Xi are deterministic. In some cases, e.g., in utility theory in economics, natural growth and convexity constraints are required on the function, which makes the inverse problem more interesting. Not only that, the data may be provided in intervals and/or measured up to an additive error. It is the purpose of this work to show how the standard method of maximum entropy, as well as the method of maximum entropy in the mean, provides an efficient method to deal with these problems.