In the article Pricing via utility maximization and entropy from Richard Rouge and Nicole El Karoui, they define the value function of the optimization problem as

\begin{align} V(x,C) = \dfrac{1}{\gamma} \ln E\left[ -\hat{U}(x,-C) \right] = \inf_{\pi \in \mathcal{A}} \dfrac{1}{\gamma} \ln E[\exp -\gamma ( X_{T}^{x, \pi} - C)], \end{align}

where $X_{t}^{x, \pi}$ is the wealth process with an initial endowment $x$ and portfolio strategy $\pi$. $C$ is the payoff of the contingent claim at time $T$. $\mathcal{A}$ is a closed convex cone, and $\hat{U}$ is the maximal expected utility function that is defined as \begin{align} \hat{U}(x,C) = \max_{\pi \in \mathcal{A}}E[U(X_{T}^{x, \pi} + C)], \end{align} where $U(y):= - \exp(-\gamma y)$, the negative exponential utility.

They use the duality between free energy and the relative entropy that is \begin{align} \ln E[\exp B] = \sup_{Q \ll P} [E^{Q}[B] -h(Q \vert P)] \end{align}

to conclude that

\begin{align} V(x,C) = \inf_{\pi \in \mathcal{A}} \sup_{Q \ll P}\left\lbrace E^{Q}[-X_{T}^{x, \pi}+C] - \dfrac{1}{\gamma}h(Q \vert P)\right\rbrace \end{align} that is the value of a stochastic game between an agent and the market.

The authors claim in Theorem 2.1 that the price of the contingent claim is given by

\begin{align} p(x,C) /B_{0,T} = \sup_{Q_{T}}\left\lbrace E^{Q_{T}}[C] - \dfrac{1}{\gamma}h(Q_{T} \vert P)\right\rbrace - \sup_{Q_{T}}\left\lbrace - \dfrac{1}{\gamma}h(Q_{T} \vert P)\right\rbrace \end{align} where $Q_{T}$ runs trough the set of probabilities $Q_{T} \sim P$. The authors claim this result holds in a general setting but they do not explicitly show how they deduce this price. Does anyone know how to deduce this result under the general setting or under the Brownian motion setting mentioned below (that appears in the article)? or any reference?

By the way, the authors define the price $p(x,C)$ of a contingent claim as the smallest $p$ such that \begin{align} \sup_{\pi \in \mathcal{A}}E[U(X_{T}^{x+p, \pi}- C)] \geq E[U(X_{T}^{x, \pi})] \end{align}

Even under the the following Brownian motion model that they provide, they do not explicitly show how to deduce the price.

\begin{align} dP_{t}^{0} =r_{t}P_{t}^{0}dt, \ \ \ P_{0}^{0}=1, \end{align} is the dynamics of the riskless asset price.

\begin{align} dP_{t}^{i} = P_{t}^{i}\left[ b_{t}^{i}dt + \sum_{j=1}^{d} \sigma_{t}^{i j} dW_{t}^{j}\right] \end{align} are the dynamics of the risky assets, where $P_{t}^{i}$ is the price of the risky ith risky asset at time $t$.

\begin{align} X_{0}^{x, \pi}=x \ \ \ \ dX_{t} = \left( X_{t} - \sum_{i=1}^{d} \pi_{t}^{i} \right)r_{t} dt + \sum_{i=1}^{d}\pi_{t}^{i}\left[ b_{t}^{i}dt + \sum_{j=1}^{d} \sigma_{t}^{i j} dW_{t}^{j}\right] \end{align} is the equation of the wealth process $X^{x, \pi}$ where $\pi_{t}^{i}$ is the amount invested in the ith risky asset with $i =1,...,d$.

and the zero coupon bond $B_{t,T}$ is used as numeraire and follow the equation \begin{align} dB_{t,T} = B_{t,T} [(r_{t} + \sigma_{t}^{T^{*}}\sigma_{t})dt + \sigma_{t}^{T^{*}} \sigma_{t} dW_{t}] \end{align} where $\sigma_{t}^{T}$ is an $\mathbb{R}^{d}$-valued progressively measurable process, and $*$ denotes transpose.

This is the only result that I have not been able to prove. Thanks in advance.


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