Relative and Discrete Utility Maximising Entropy

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 arbitra...

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Bibliographic Details
Published in:arXiv.org 2007-09
Main Authors: Harańczyk, Grzegorz, Słomczyński, Wojciech, Zastawniak, Tomasz
Format: Article
Language:English
Subjects:
Entropy
Entropy (Information theory)
Expected utility
Maximization
Optimization
Online Access:Get full text
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Summary: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.
ISSN:2331-8422