Optimal portfolio for an insider in a market driven by Levy processes

We consider a financial market driven by a Levy process with filtration  [image omitted]. An insider in this market is an agent who has access to more information than an honest trader. Mathematically, this is modelled by allowing a strategy of an insider to be adapted to a bigger filtration  [image omitted]. The corresponding anticipating stochastic differential equation of the wealth is interpreted in the sense of forward integrals. In this framework, we study the optimal portfolio problem of an insider with logarithmic utility function. Explicit results are given in the case where the jumps are generated by a Poisson process.