  We consider the mean--variance portfolio optimization problem under the game
theoretic framework and without risk-free assets. The problem is solved
semi-explicitly by applying the extended Hamilton--Jacobi--Bellman equation.
Although the coefficient of risk aversion in our model is a constant, the
optimal amounts of money invested in each stock still depend on the current
wealth in general. The optimal solution is obtained by solving a system of
ordinary differential equations whose existence and uniqueness are proved and a
numerical algorithm as well as its convergence speed are provided. Different
from portfolio selection with risk-free assets, our value function is quadratic
in the current wealth, and the equilibrium allocation is linearly sensitive to
the initial wealth. Numerical results show that this model performs better than
both the classical one and the variance model in a bull market.
