The minimal entropy martingale measure and the valuation problem in incomplete markets

Let $\chi $ be a family of stochastic processes on a given filtered probability space $(\Omega ,\mathcal{F},(\mathcal{F}_t)_{t\in \mathcal{T}%},P) $ with $\mathcal{T}\subseteq \mathbb{R}_{+}$. Under the assumption that the set $\mathcal{M}_e$ of equivalent martingale measures for $\chi $ is not empty, we give sufficient conditions for the existence of a unique equivalent martingale measure which minimizes the relative entropy, with respect to $P,$ in the class of martingale measures. We then provide the characterization of the density of the Minimal Entropy Martingale measure, which suggests the equivalence between the maximization of expected exponential utility and the minimization of the relative entropy.