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.All sources of funding of the study should be disclosed. Please clearly indicate grants that you have received in support of your research work. Clearly state if you received funds for covering the costs to publish in open access

Maxentropic approach to decompound aggregate risk losses

A risk manager may be faced with the following problem: she/he has obtained loss data collected during a year, but the data only contains the total number of events and the total loss for that year. She/he suspects that there are different sources of risk, each occurring with a different frequency, and wants to identify the frequency with which each type of event occurs and if possible, the individual losses at each risk event. The purpose of this methodological note is to examine a combination of disentangling and decompounding procedures, to get as close as possible to that goal. The disentangling procedure is actually a two step process: First, a preliminary analysis is carried out to determine the number of risks groups present. Once that is decided, the underlying model for the frequency of each type of risk is worked out. After that we use the maxentropic techniques in the decompounding stage to determine the distribution of individual losses that aggregated yield the observed total los

Two maxentropic approaches to determine the probability density of compound risk losses

Here we present an application of two maxentropic procedures to determine the probability density distribution of a compound random variable describing aggregate risk, using only a finite number of empirically determined fractional moments. The two methods that we use are the Standard method of Maximum Entropy (SME) and the method of Maximum Entropy in the Mean (MEM). We analyze the performance and robustness of these two procedures in several numerical examples, in which the frequency of losses is Poisson and the individual losses are lognormal random variables. We shall verify that the reconstructions obtained pass a variety of statistical quality criteria, and provide good estimations of VaR and TVaR, which are important measures for risk management purposes. As side product of the work, we obtain a rather accurate numerical description of the density of such compound random variable. These approaches are also used to develop a procedure to determine the distribution of the individual losses from the knowledge of the total loss. Thus, if the only information available is the total loss, and the nature of the frequency of losses is known, the method of maximum entropy provides an efficient method to determine the individual losses as wel