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

We consider a non-consuming agent interested in the maximization of the long-run growth rate of a wealth process investing either in a money market and in one risky asset following a geometric Brownian motion or in futures following an arithmetic Brownian motion. The agent faces proportional transaction costs, and similarly as in [17] where the case of stock trading is considered, we show how the log-optimal optimal policies in the long run can be derived when using the technical tool of shadow prices. We also provide a brief link between technical tools used in this paper and the ones used in [14,15,17].