Optimal consumption and sale strategies for a risk averse agent (Q2832613)

The paper is concerned with the optimal behavior of an agent whose goal is to maximize the expected discounted utility of consumption, and who finances consumption from a combination of initial wealth and from the sale of an initial endowment of an infinitely divisible security. Her actions are to choose an optimal consumption strategy and an optimal holding or portfolio of a risky security, under the restriction that the risky asset can only be sold, and purchases are not permitted. The authors consider a special case of the transaction cost model in which the transaction costs associated with purchases of the risky asset are infinite. Effectively purchases are disallowed, and one may think of an agent who is endowed with a quantity of an asset which she may sell, but which she may not trade dynamically. As to the main tools, the authors take the classical stochastic control approach to the primal problem. These methods arguably lead to a simpler set of governing equations than those that arise from the shadow price method. The HJB equation for the problem is of second order, nonlinear, and subject to value matching and smooth fit of the first and second derivatives at an unknown free boundary. It is shown that the problem can be reduced to a crossing problem for the solution of a first-order ODE. In the case of a finite and positive critical ratio it is demonstrated how the solution to the problem can be characterized by an autonomous one-dimensional diffusion process with reflection and its local time.