Dynamic Optimization for a Mixed Portfolio with Transaction Costs

Summary

Introduction

We consider a model of n risky and one risk-free asset in which the dynamics of the prices of the risky assets are governed by logarithmic Brownian motions (LBM). Risky assets are conventionally called Stocks while the risk-free asset is called a Bond or Bank. Modelling the stock prices by LBM goes back to Black & Scholes (1973). Since the paper by Merton (1971), in which a dynamic optimization consumption/investment model was introduced, there has been a lot of effort expanded to develop and perfect the LBM models.

One of the avenues of research deals with the transaction cost problem: Consider an investor who has an initial wealth invested in Stocks and Bonds and who has the ability to transfer funds between the assets. This transfer involves brokerage fees (transaction costs).

In one type of model the proceeds from these operations are used to finance the consumption and the objective is to maximize the cumulative expected utility of consumption over a finite or infinite time horizon (see Davis & Norman 1990, Akian, Menaldi & Sulem 1996, Fleming & Soner 1992, Shreve & Soner 1994, Zariphopoulou 1989). In Akian, Sulem & Séquier (1995), a similar problem is considered but on a finite horizon and without consumption. In Sethi & Taksar (1988) and Akian, Sulem & Taksar (1996), the objective is to maximize the long run average growth of wealth.

Mathematicaly, these problems can be formulated as singular stochastic control problems.