  This survey is an introduction to asymptotic methods for portfolio-choice
problems with small transaction costs. We outline how to derive the
corresponding dynamic programming equations and simplify them in the small-cost
limit. This allows to obtain explicit solutions in a wide range of settings,
which we illustrate for a model with mean-reverting expected returns and
proportional transaction costs. For even more complex models, we present a
policy iteration scheme that allows to compute the solution numerically.
