Shadow Prices and Well-Posedness in the Problem of Optimal Investment and Consumption with Transaction Costs

We revisit the optimal investment and consumption model of Davis and Norman [Math. Oper. Res., 15 (1990), pp. 676--713] and Shreve and Soner [Ann. Appl. Probab., 4 (1994), pp. 609--692], following a shadow-price approach similar to that of Kallsen and Muhle-Karbe [Ann. Appl. Probab., 20 (2010), pp. 1341--1358]. Making use of the completeness of the model without transaction costs, we reformulate and reduce the Hamilton--Jacobi--Bellman equation for this singular stochastic control problem to a nonstandard free-boundary problem for a first-order ODE with an integral constraint. Having shown that the free boundary problem has a smooth solution, we use it to construct the solution of the original optimal investment/consumption problem in a self-contained manner and without any recourse to the dynamic programming principle. Furthermore, we provide an explicit characterization of model parameters for which the value function is finite.