Utility Maximization on the Real Line under Proportional Transaction Costs

We consider a financial market with costs as in Kabanov and Last (1999). Given a utility function defined on ${\mathbb R}$, we analyze the problem of maximizing the expected utility of the liquidation value of terminal wealth diminished by some random claim. We prove that, under the Reasonable asymptotic elasticity conditions introduced by Schachermayer (2000a), existence and duality hold in the class of targets that can be approximated by bounded from below strategies. Under some additional condition, we prove that the optimal target is indeed attainable. As an application, we obtain a dual formulation for the exponential reservation price.