Utility Functions

Abstract

This chapter studies an investor’s utility function. We start with a normalized market $\left (S,(\mathscr {F}_{t}),\mathbb {P}\right )$ where the money market account’s (mma’s) value is B _t  ≡ 1. For the chapters in Part I of this book, although unstated, we implicitly assumed that the trader’s beliefs were equivalent to the statistical probability measure $\mathbb {P}$, i.e. the trader’s beliefs and the statistical probability measure agree on zero probability events. For this part of the book, Part II, we let the probability measure $\mathbb {P}$ correspond to the trader’s beliefs. This should cause no confusion since we do not need additional notation for the statistical probability measure in this part of the book. We discuss differential beliefs in Sect. 9.7 below. In addition, consistent with this interpretation, we let the information filtration $\mathscr {F}_{t}$ given above correspond to the trader’s information set. When we study the notion of an equilibrium in Part III of this book, we will introduce a distinction between the trader’s beliefs and the statistical probability measure, and a distinction between the trader’s information set and the market’s information set.