This paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate 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. Existence and uniqueness of a viscosity solution are proved. The variational inequality is solved by using a numerical algorithm based on policies, iterations, and multigrid methods. Numerical results are displayed for n = 1 and n = 2.
Numerical Analysis and Computation | Probability
M. Akian, J.-L. Menaldi and A. Sulem, On an investment-consumption model with transaction costs, SIAM J. Control Optim., 34 (1996), pp. 329-364. doi: 10.1137/S0363012993247159