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

Abstract

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.

Construction of Discrete Time Shadow Price

“…The method used in this paper is significantly different from those considered in the other papers (see [1,2,5,6,9,10,13,15,16,19]). Because of discrete time we don't have differential structure of the model as in [15].…”

Log-optimal investment in the long run with proportional transaction costs when using shadow prices

“…• in shadow prices is referred to [4,5,6,7,16,17,18,19,24,25,26,32], • in logarithmic utility to [1,3,8,9,10,11,20,27,33,34,39,40], • in the investment-consumption problem with presence of proportional transaction costs to [2,3,12,21,22,28,29,30,36].…”

“…By the self-financing condition (6,16) and the definition of (W (9,15), we obtain the equality on the left-hand side in the following…”

Worst-Case Portfolio Optimization with Proportional Transaction Costs

“…The interested reader is referred to e.g. Kallsen and Muhle-Karbe [17], Gerhold et al [14], Choi et al [7] and the references therein for a different approach to tackle the problem.…”