  In this paper we study an optimal portfolio selection problem under
instantaneous price impact. Based on some empirical analysis in the literature,
we model such impact as a concave function of the trading size when the trading
size is small. The price impact can be thought of as either a liquidity cost or
a transaction cost, but the concavity nature of the cost leads to some
fundamental difference from those in the existing literature. We show that the
problem can be reduced to an impulse control problem, but without fixed cost,
and that the value function is a viscosity solution to a special type of
Quasi-Variational Inequality (QVI). We also prove directly (without using the
solution to the QVI) that the optimal strategy exists and more importantly,
despite the absence of a fixed cost, it is still in a "piecewise constant"
form, reflecting a more practical perspective.
