On a free boundary problem that arises in portfolio management

We study a model for the optimal management of a portfolio when there are transaction costs proportional to a fixed fraction of the portfolio value. The risky securities are modelled as correlated geometric brownian motions. There is a riskless bank account and the aim is to maximize the long-run growth rate. It is known that the optimal trading strategy is characterized by the solution of a certain partial differential equation free boundary problem. This paper explains how to transform this free boundary problem for the case of three securities into a much simpler one that is feasible to solve with numerical methods.