Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs

TL;DR

This paper revisits the optimal investment and consumption problem with transaction costs, using a shadow-price approach to reformulate the problem into a free-boundary ODE, providing explicit conditions for the finiteness of the value function.

Contribution

It introduces a novel reduction of the Hamilton-Jacobi-Bellman equation to a free-boundary problem and constructs solutions without relying on dynamic programming.

Findings

Successfully reduces the problem to a free-boundary ODE with an integral constraint.

Provides explicit parameter conditions ensuring the value function's finiteness.

Constructs solutions using the free-boundary problem without dynamic programming.

Abstract

We revisit the optimal investment and consumption model of Davis and Norman (1990) and Shreve and Soner (1994), following a shadow-price approach similar to that of Kallsen and Muhle-Karbe (2010). 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 non-standard 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.