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 bounded from below stochastic integrals 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 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 H^W. We show that, for all loss variables W contained in a properly identified set W, the optimal value on the class H^W is constant and coincides with the optimal value of the maximization problem over a larger domain K_Φ. The class K_Φ doesn't depend on the single W∈W, but it depends on the utility function u through its conjugate function Φ. By duality methods we show that the optimal solution exists in K_Φ and it can be represented as a stochastic integral that is a uniformly integrable martingale under the minimax measure. We provide the economic interpretation of the larger class K_Φ and we analyze some examples that show that this enlargement of the class of trading strategies is indeed necessary.