Relative and Discrete Utility Maximising Entropy

Grzegorz Haranczyk; Wojciech Slomczynski; Tomasz Zastawniak

doi:10.1142/S1230161208000213

Relative and Discrete Utility Maximising Entropy

Grzegorz Haranczyk, Wojciech Slomczynski, Tomasz Zastawniak

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The notion of utility maximising entropy (u-entropy)of a probability density, which was introduced and studied in[37], 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 logarithmicor isoelastic(a powerfunction), then the well-known notions of Boltzmann-Shannon andRenyi 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 lierature.

Original language	English
Pages (from-to)	303-327
Number of pages	25
Journal	Open systems & information dynamics
Volume	15
Issue number	4
DOIs	https://doi.org/10.1142/S1230161208000213
Publication status	Published - Dec 2008

Keywords

INCOMPLETE MARKETS
OPTIMAL INVESTMENT
MAXIMIZATION

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10.1142/S1230161208000213

http://arxiv.org/abs/0709.1281

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abstract = "The notion of utility maximising entropy (u-entropy)of a probability density, which was introduced and studied in[37], 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 logarithmicor isoelastic(a powerfunction), then the well-known notions of Boltzmann-Shannon andRenyi 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 lierature.",

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T1 - Relative and Discrete Utility Maximising Entropy

AU - Haranczyk, Grzegorz

AU - Slomczynski, Wojciech

AU - Zastawniak, Tomasz

PY - 2008/12

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N2 - The notion of utility maximising entropy (u-entropy)of a probability density, which was introduced and studied in[37], 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 logarithmicor isoelastic(a powerfunction), then the well-known notions of Boltzmann-Shannon andRenyi 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 lierature.

AB - The notion of utility maximising entropy (u-entropy)of a probability density, which was introduced and studied in[37], 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 logarithmicor isoelastic(a powerfunction), then the well-known notions of Boltzmann-Shannon andRenyi 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 lierature.

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KW - OPTIMAL INVESTMENT

KW - MAXIMIZATION

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Relative and Discrete Utility Maximising Entropy

Abstract

Keywords

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