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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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    1. Marco Frittelli, 2000. "Introduction to a theory of value coherent with the no-arbitrage principle," Finance and Stochastics, Springer, vol. 4(3), pages 275-297.
    2. Amendinger, Jürgen & Imkeller, Peter & Schweizer, Martin, 1998. "Additional logarithmic utility of an insider," Stochastic Processes and their Applications, Elsevier, vol. 75(2), pages 263-286, July.
    3. Fabio Bellini & Marco Frittelli, 2002. "On the Existence of Minimax Martingale Measures," Mathematical Finance, Wiley Blackwell, vol. 12(1), pages 1-21.
    4. Thomas Goll & Ludger Rüschendorf, 2001. "Minimax and minimal distance martingale measures and their relationship to portfolio optimization," Finance and Stochastics, Springer, vol. 5(4), pages 557-581.
    5. Amendinger, Jürgen & Imkeller, Peter & Schweizer, Martin, 1998. "Additional logarithmic utility of an insider," SFB 373 Discussion Papers 1998,25, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    6. Anne Gundel, 2005. "Robust utility maximization for complete and incomplete market models," Finance and Stochastics, Springer, vol. 9(2), pages 151-176, April.
    7. Sara Biagini & Marco Frittelli, 2005. "Utility maximization in incomplete markets for unbounded processes," Finance and Stochastics, Springer, vol. 9(4), pages 493-517, October.
    8. Ying Hu & Peter Imkeller & Matthias Muller, 2005. "Utility maximization in incomplete markets," Papers math/0508448, arXiv.org.
    9. Richard Rouge & Nicole El Karoui, 2000. "Pricing Via Utility Maximization and Entropy," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 259-276.
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