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 1971, many variants of the original problem have been put forward and extensively studied. The variant we consider here is the Merton problem with a drawdown constraint on consumption. That is, the consumption can never fall below a fixed proportion of the running maximum of past consumption. In terms of economic motivation, this constraint represents a type of habit formation where once an investor has reached a certain standard of living, he is reluctant to let his standard of living fall too far below that level.

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 – subject to the drawdown constraint on consumption.

We consider power utility functions and use techniques from stochastic optimal control to identify a candidate solution. Finally, we discuss how to verify our conjectured solution.