  We reconsider the problem of optimal trading in the presence of linear and
quadratic costs, for arbitrary linear costs but in the limit where quadratic
costs are small. Using matched asymptotic expansion techniques, we find that
the trading speed vanishes inside a band that is narrower than in the absence
of quadratic costs, by an amount that scales as the one-third power of
quadratic costs. Outside of the band, we find three regimes: a small boundary
layer where the velocity vanishes linearly with the distance to the band, an
intermediate region where the velocity behaves as a square-root of that
distance, and a far region where it becomes linear. Our solution is consistent
with available numerical results. We determine the conditions in which our
expansion is useful in practical applications, and generalize our solution to
other forms of non-linear costs.
