Optimal investment in a Lévy market

The stock price process is modelled by a geometric Lévy process (tak-ing into account jumps). Except for the geometric Brownian model and the geometric Poissonian model, the resulting models are incomplete and there are many equivalent martingale measures. However the model can be completed by the so called power-jump assets. By doing this we allow investment in these new assets and we can try to maximize the utility of these portfolios. As particular cases we obtain the optimal portfolios based in stocks and bonds, showing that the new assets are superfluous for certain martingale measures that depend on the utility function we use