Consumption–Investment Problems

Abstract

The study of consumption–investment problems in continuous time was initiated by Merton. He considered a model of frictionless market where the price processes are geometric Brownian motions and the investor’s goal is to maximize the expected discounted utility of consumption on the infinite time interval. For the power utility function, he obtained an explicit solution of the optimal control problem. This solution has a clear financial meaning: the optimal investment is to keep the proportions of the total wealth held in risky securities equal to a constant vector. The latter is easily calculated from the model parameters. This work was extended by many authors in various directions including models with transaction costs, which are the main objects of our interest. Taking into account that the Merton problem is classical and exposed in a number of textbooks, we give here a rather sketchy presentation needed to understand basic ideas and methods as well as their evolution. The results of this section will be used at the end of this chapter, where we discuss an asymptotical behavior of the consumption–investment problem for small transaction cost coefficients.