Optimal Investment in a Levy Market

In this paper we consider the optimal investment problem in a market where the stock price process is modeled by a geometric Levy process (taking 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 expected 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.