Integral Equations and Phase Transitions in Stochastic Games. An Analogy with Statistical Physics

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.