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I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure $Q$ with respect to the probability measure $P$ defined by $h(Q \vert P) := E[dQ/dP \ln(dQ/dP)]$ if $Q \ll P$, $+\infty$ else. They also talk about the concept of free energy of a random variable $B$, and this is equal to $\ln E[\exp B]$. They claim that for a bounded random variable $B$, entropy and free energy are in relation by the following duality:

\begin{align} \ln E[\exp B] = \sup_{Q \ll P} [ E^{Q}[B] - h(Q \vert P) ] \end{align}

Does anyone know an article that shows the proof? or does anyone know how to deduce this equation?

The interesting feature is that with this formula we can deduce the stochastic game between an agent and the market.

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Such relationships are commonly covered in statistical mechanics, so any decent statsictal mechanics book should help. Here is an article that gives a very nice summary of the key concepts:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2716075/#!po=4.74138

I don’t have access to Rouge and Karoui article, but I think it is just trying to find an equivalent Martingale measure that maximises the free energy (think of it as a measure of sort of stability).

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