Abstract: We first consider the problem of representation of the Q-martingale \\wt\Lambda_t :=\ep^Q\Lambda_T|\cal G_t\. Then we consider a market of a stock price affected by a stochastic factor, in which there exists a insider who only knows the price information and a side information. We consider his problem of optimal utility for terminal wealth with and without side-information, and obtain a form of optimal terminal wealth in two cases. Finally, we compare these two cases for the logarithmic utility, and analyze the influence of the `side information'.