Relative and Discrete Utility Maximising Entropy

TL;DR

This paper extends the concept of utility maximising entropy (u-entropy) to relative and discrete cases, connecting it with thermodynamics and information theory, and establishing its fundamental properties and relations.

Contribution

It introduces the relative and absolute u-entropy for probability measures, generalizing previous work and linking it to thermodynamics and financial utility maximization.

Findings

Defined relative u-entropy for arbitrary probability spaces

Introduced absolute u-entropy for discrete probability spaces

Revealed connections with Boltzmann-Shannon and Renyi entropies

Abstract

The notion of utility maximising entropy (u-entropy) of a probability density, which was introduced and studied by Slomczynski and Zastawniak (Ann. Prob 32 (2004) 2261-2285, arXiv:math.PR/0410115 v1), is extended in two directions. First, the relative u-entropy of two probability measures in arbitrary probability spaces is defined. Then, specialising to discrete probability spaces, we also introduce the absolute u-entropy of a probability measure. Both notions are based on the idea, borrowed from mathematical finance, of maximising the expected utility of the terminal wealth of an investor. Moreover, u-entropy is also relevant in thermodynamics, as it can replace the standard Boltzmann-Shannon entropy in the Second Law. If the utility function is logarithmic or isoelastic (a power function), then the well-known notions of the Boltzmann-Shannon and Renyi relative entropy are recovered. We establish the principal properties of relative and discrete u-entropy and discuss the links with several related approaches in the literature.