Abstract

The Merton problem – a question about optimal portfolio selection and consumption in continuous time – is indeed ubiquitous throughout the mathematical finance literature. Since Merton’s seminal paper in 1969, many variants of the original problem have been put forward and extensively studied. First we will consider the standard Merton problem. To be precise, we consider an agent who can invest in a risk-free asset and a risky stock modelled by geometric Brownian motion. The agent seeks to maximise the expected infinite horizon utility of consumption by finding the optimal portfolio selection and consumption strategies. We work with power utility functions because they enable us to construct explicit solutions. If time permits, we will also consider a variant of the original problem where we impose a drawdown constraint on consumption. That is, the consumption can never fall below a fixed proportion of the running maximum of past consumption.