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 L p -semimartingale setting. In particular, we solve the problem for utility functions of type − e − x (exponential problem) and − ( 1 − x 2 m ) 2 m ( 2 m -th problem). The convergence of the 2 m -th problems to the exponential one is proved. Using this result an explicit portfolio for the exponential problem is derived.