Appl Math Optim 54:95–116 (2006)
2006 Springer Science+Business Media, Inc.
Maximizing the Growth Rate of a Portfolio with Fixed
and Proportional Transaction Costs
Department of Systems Innovation, Graduate School of Engineering Science,
Osaka University, Toyonaka 560-8531, Japan
Abstract. In this paper a portfolio optimization problem with transaction costs is
studied. Transactions of assets are formulated as an impulsive control, which does
not allow continuous transactions. The introduction of ﬁxed rate costs has the effect
of preventing continuous transactions. The objective of this paper is studying the
problem of maximizing the growth rate of expected log utility. A quasi-variational
inequality (QVI) of “ergodic type” is derived from the optimization problem. To
solve the inequality, we use a perturbation method, where we obtain a necessary
estimate of solutions of non-ergodic type QVIs by using a stochastic representation
of the solutions.
Key Words. Portfolio optimization, Transaction costs, Quasi-variational inequal-
ities, Impulsive stochastic control.
AMS Classiﬁcation. 91B20, 93E20, 49N25.
This paper concerns a portfolio optimization problem with transaction costs. We for-
mulate the optimization problem as an impulsive control problem in which ﬁxed and
proportional transaction costs are introduced, and continuous trading is not allowed. The
objective of this paper is studying the problem of maximizing the growth rate of expected
Portfolio optimization problems with transaction costs have been widely studied by
various authors, e.g., , , –, and . Especially, Taksar et al.  formulate
portfolio optimization as a singular control problem and study the same maximization
problem as ours. Their study is extended by Akian et al.  to a multi-dimensional