  This paper studies the continuous time mean-variance portfolio selection
problem with one kind of non-linear wealth dynamics. To deal the expectation
constraint, an auxiliary stochastic control problem is firstly solved by two
new generalized stochastic Riccati equations from which a candidate portfolio
in feedback form is constructed, and the corresponding wealth process will
never cross the vertex of the parabola. In order to verify the optimality of
the candidate portfolio, the convex duality (requires the monotonicity of the
cost function) is established to give another more direct expression of the
terminal wealth level. The variance-optimal martingale measure and the link
between the non-linear financial market and the classical linear market are
also provided. Finally, we obtain the efficient frontier in closed form. From
our results, people are more likely to invest their money in riskless asset
compared with the classical linear market.
