Asymptotics and Duality for the Davis and Norman Problem

We revisit the problem of maximizing expected logarithmic utility from consumption over an infinite horizon in the Black-Scholes model with proportional transaction costs, as studied in the seminal paper of Davis and Norman [Math. Operation Research, 15, 1990]. Similarly to Kallsen and Muhle-Karbe [Ann. Appl. Probab., 20, 2010], we tackle this problem by determining a shadow price, that is, a frictionless price process with values in the bid-ask spread which leads to the same optimization problem. However, we use a different parametrization, which facilitates computation and verification. Moreover, for small transaction costs, we determine fractional Taylor expansions of arbitrary order for the boundaries of the no-trade region and the value function. This extends work of Janecek and Shreve [Finance Stoch., 8, 2004], who determined the leading terms of these power series.