  We present an expansion for portfolio optimization in the presence of small,
instantaneous, quadratic transaction costs. Specifically, the magnitude of
transaction costs has a coefficient that is of the order $\epsilon$ small,
which leads to the optimization problem having an asymptotically-singular
Hamilton-Jacobi-Bellman equation whose solution can be expanded in powers of
$\sqrt\epsilon$. In this paper we derive explicit formulae for the first two
terms of this expansion. Analysis and simulation are provided to show the
behavior of this approximating solution.
