  In this paper, we consider the stochastic optimal control problems under
model risk caused by uncertain volatilities. To have a mathematical consistent
framework we use the notion of G-expectation and its corresponding G-Brwonian
motion introduced by Peng(2007). Based on the theory of stochastic differential
equations on a sublinear expectation space
$(\Omega,\mathcal{H},\hat{\mathbb{E}})$, we prove a stochastic maximum
principle for controlled processes driven by G-Brownian motion. Then we obtain
the maximum condition in terms of the $\mathcal{H}$-function plus some
convexity conditions constitute sufficient conditions for optimality. Finally,
we solve a portfolio optimization problem with ambiguous volatility as an
explicitly illustrated example of the main result.
