On convergence to the exponential utility problem

We provide a method for solving dynamic expected utility maximization problems with possibly not everywhere increasing utility functions in an LpLp-semimartingale setting. In particular, we solve the problem for utility functions of type −e−x (exponential problem) and −(1−x2m)2m (2m2m-th problem). The convergence of the 2m2m-th problems to the exponential one is proved. Using this result an explicit portfolio for the exponential problem is derived.