A Stochastic Maximum Principle for Processes Driven by G-Brownian Motion and Applications to Finance

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.