  In this paper, we study representative investor's G-utility maximization
problem by G-martingale approach in the framework of G-expectation space
proposed by Peng \cite{Pe19}. Financial market has only a bond and a stock with
uncertainty characterized by G-Brownian motions. The routine idea of \cite{Wxz}
fails because that the quadratic variation process of a G-Brownian motion is
also a stochastic process. To overcome this difficulty, an extended nonlinear
expectation should be pulled in. A sufficient condition of G-utility
maximization is presented firstly. In the case of log-utility, an explicit
solution of optimal strategy can be obtained by constructing and solving a
couple of G-FBSDEs, then verifying the optimal strategy to meet the sufficient
condition. As an application, an explicit solution of a stochastic interest
model is obtained by the same approach. All economic meanings of optimal
strategies are consistent with our intuitions.
