Multi-asset portfolio selection problem with transaction costs

This paper considers the optimal consumption and investiment policy for an investor who has available one bank account paying a fixed interest rate r and n risky assets whose prices are log-normal diffusions. We suppose that transactions between the assets incur a cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption. Dynamic Programming leads to a Variational Inequality for the value function which is solved by using a numerical algorithm based on policies iterations and multigrid methods. Numerical results are displayed for n = 1 and n = 2.