Finite horizon multidimensional portfolio selection problem with singular transactions

This paper considers the optimal investment policy for an investor who has available one bank account paying a fixed interest rate r and n risky assets whose prices are correlated 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 a function of the total net wealth on a finite horizon. Dynamic programming leads to a parabolic variational inequality for the value function which is solved by using a numerical algorithm based on policies iterations and multigrid methods. Numerical results are presented dealing with the issue of domestic asset allocation, that is the optimal split between cash, long bonds and equities. The impact of the transaction costs on the risk return characteristics of the optimal policies is analyzed.