Geodesic
Integral Equations and Phase Transitions in Stochastic Games. An Analogy with Statistical Physics
Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 2, pp. 403-411

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Maximization of the Kullback–Leibler information is known to result in general Esscher transformations. The Bose–Einstein and Fermi–Dirac statistics in a probability space $(\Omega, \mathcal{F},P)$ give rise to another kind of information, namely, $$ S_B=\int \log\bigg(1+\frac{dP}{dQ}\bigg)\,dQ+ \int \log\bigg(1+\frac{dQ}{dP}\bigg)\,dP $$ for the Bose statistics and $$ S_F =\int\log\bigg(\frac{dP}{dQ}-1\bigg)\,dQ -\int\log\bigg(1-\frac{dQ}{dP}\bigg)\,dP, \qquad \frac{dP}{dQ} >1, $$ for the Fermi statistics. This information generates measure transformations corresponding to these statistics. In the presence of a payoff matrix, these transformations vary in accordance with the integral equations given in the paper. We give examples of financial games corresponding to Bose and Fermi statistics.
Keywords: Bose statistics, Fermi statistics, payoff matrix, entropy, integral equation, Kullback–Leibler information, thermodynamics, statistical physics, dyadic games.
Mots-clés : Esscher transformation, phase transition 
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     author = {V. P. Maslov},
     title = {Integral {Equations} and {Phase} {Transitions} in {Stochastic} {Games.} {An} {Analogy} with {Statistical} {Physics}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {403--411},
     publisher = {mathdoc},
     volume = {48},
     number = {2},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2003_48_2_a12/}
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V. P. Maslov. Integral Equations and Phase Transitions in Stochastic Games. An Analogy with Statistical Physics. Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 2, pp. 403-411. http://geodesic.mathdoc.fr/item/TVP_2003_48_2_a12/