  We apply numerical dynamic programming techniques to solve discrete-time
multi-asset dynamic portfolio optimization problems with proportional
transaction costs and shorting/borrowing constraints. Examples include problems
with multiple assets, and many trading periods in a finite horizon problem. We
also solve dynamic stochastic problems, with a portfolio including one
risk-free asset, an option, and its underlying risky asset, under the existence
of transaction costs and constraints. These examples show that it is now
tractable to solve such problems.
