Discrete Time Growth Optimal Investment With Costs

Abstract

In this work we ask how should an investor distribute wealth over various assets to maximize the growth rate of the cumulative wealth in a discrete time market with proportional transaction costs. We show that this sequential decision problem has a stationary optimal policy. In addition, we show that for all ε > 0 there exists a policy that guarantees a growth rate at most ε below optimal on almost every sample path. We also show the existence of an ε-optimal control- limit policies – control-limit policies correct the portfolio only it leaves a compact connected no-trade set. For the special case of two-asset markets, we establish that for all ε > 0 there exists a control-limit policy that is ε-optimal with probability 1.

Keywords: Growth optimal investment, transaction costs, Markov decision processes

JEL Classification: G11