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Relative and Discrete Utility Maximising Entropy

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  • Grzegorz Hara'nczyk
  • Wojciech S{l}omczy'nski
  • Tomasz Zastawniak

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.

Suggested Citation

  • Grzegorz Hara'nczyk & Wojciech S{l}omczy'nski & Tomasz Zastawniak, 2007. "Relative and Discrete Utility Maximising Entropy," Papers 0709.1281, arXiv.org.
    • Handle: RePEc:arx:papers:0709.1281
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    References listed on IDEAS

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