Utility maximization in an insider influenced market

We study a controlled stochastic system whose state is described by a stochastic differential equation where the coefficients are anticipating. This setting is used to interpret markets where insiders have some influence on the dynamics of prices. We give a characterization theorem for the optimal logarithmic portfolio of an investor with a different information flow from that of the insider. As examples, we provide explicit results in the partial information case which we extend in order to incorporate the enlargement of filtration techniques for markets with insiders. Finally, we consider a market with an insider which influences the drift of the asset process. This last example, which does not seem to fit into the enlargement of filtration set-up, gives a situation where it makes a difference for a small agent to acknowledge the existence of the insider in the market