RESUMEN

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.

Using fractional calculus we define integrals of the form , where and are vector-valued Hölder continuous functions of order and is a continuously differentiable function such that is -Hölder continuous for some . Under some further smooth conditions on the integral is a continuous functional of , , and the tensor product with respect to the Hölder norms. We derive some estimates for these integrals and we solve differential equations driven by the function . We discuss some applications to stochastic integrals and stochastic differential equations