Kevin Pettersson: Stochastics and Its Application in Merton’s Problem

Abstract.

In this thesis, we aim to present key concepts in stochastic processes, Itô’s calculus, and stochastic dierential equations. We will give a brief overview of how stochastic dierential equations can be used in stochastic dynamic con- trol problems and how to find explicit solutions to such optimization problems. Our main focus will be to provide explicit solutions for the optimal consump- tion and investment rules in the case when the risk aversion is constant. Lastly, we will also provide concise economic interpretations and implications of such optimal consumption and investment rules.