The Merton Problem and the Methods of Optimal Stochastic Control

The Merton Problem and the Methods of Optimal Stochastic Control

Statistics

Alex Irons

2023

We consider the infinite-horizon Merton investment-consumption problem. We begin by formally introducing the problem in the simplest, one-dimensional setting and show how one might derive a candidate solution. We then detail two different rigorous verification arguments showing this strategy is indeed the optimal one. After this, we move to a slightly more heuristic setting where we will consider two different variations of the problem - namely the case of transaction costs and non-constant parameters - and discuss to what extents the previous arguments extend to these more general situations. The report is written to be a relatively gentle introduction to the field of optimal stochastic control and continuous time finance, and to be accessible to any student with some reasonable prior understanding of stochastic calculus.