Utility Maximization Under G-Expectation

In this paper, optimal investment strategies are formulated to maximize the nonlinear expected utility of small traders, who invest assets in a risk-free bond and several risky stocks in finite trading interval [0, ]. Here we consider the optimization problem under the financial market with and without constrains respectively.We use exponential utility function as our utility function and assume nonlinear expectation based on -expectation. Applying the theories about Backward Stochastic Differential Equations(BSDE) to utility maximization by constructing a stochastic family, we take advantage of the comparison theorem and-martingale.Under the market without constrains, the existence of the optimal trading strategy is proved and the formula is worked out when we take = 2and = 2+ .Under the market with constrains, we assume the constraints of trading strategies to be closed sets. In the first part we further assume the optimal trading strategy exists then we come to the expression. In order to prove the existence, we bring forward the concept of *-expectation for the first time.