Utility maximization in incomplete markets for unbounded processes

When the price processes of the financial assets are described by possibly unbounded semimartingales, the classical concept of admissible trading strategies may lead to a trivial utility maximization problem because the set of stochastic integrals bounded from below may be reduced to the zero process. However, it could happen that the investor is willing to trade in such a risky market, where potential losses are unlimited, in order to increase his/her expected utility. We translate this attitude into mathematical terms by employing a class $\mathcal{H}^{W}$ of W-admissible trading strategies which depend on a loss random variable W. These strategies enjoy good mathematical properties and the losses they could generate in trading are compatible with the preferences of the agent. We formulate and analyze by duality methods the utility maximization problem on the new domain $\mathcal{H}^{W}$ . We show that, for all loss variables W contained in a properly identified set $\mathcal{W}$ , the optimal value on the class $\mathcal{H}^{W}$ is constant and coincides with the optimal value of the maximization problem over a larger domain ${K} _{\Phi}.$ The class ${K}_{\Phi}$ does not depend on a single $W\in \mathcal{W},$ but it depends on the utility function u through its conjugate function $\Phi $ . By duality methods we show that the solution exists in ${K}_{\Phi}$ and can be represented as a stochastic integral that is a uniformly integrable martingale under the minimax measure. We provide an economic interpretation of the larger class ${K}_{\Phi}$ and analyze some examples to show that this enlargement of the class of trading strategies is indeed necessary. Copyright Springer-Verlag Berlin/Heidelberg 2005