In an arbitrage free incomplete market we consider the problem of maximizing terminal isoelastic utility. The relationship between the optimal portfolio, the optimal martingale measure in the dual problem and the optimal value function of the problem is described by an BSDE. For a totally unhedgeable price for instantaneous risk, isoelastic utility of terminal wealth can be maximized using a portfolio consisting of the locally risk-free bond and a locally efficient fund only. In a markovian market model we find a non-linear PDE for the logarithm of the value function. From the solution we can construct the optimal portfolio and the solution of the dual problem.

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