Problems and Solutions in Mathematical Finance: Stochastic Calculus, Volume I

Chapter 1General Probability Theory

Probability theory is a branch of mathematics that deals with mathematical models of trials whose outcomes depend on chance. Within the context of mathematical finance, we will review some basic concepts of probability theory that are needed to begin solving stochastic calculus problems. The topics covered in this chapter are by no means exhaustive but are sufficient to be utilised in the following chapters and in later volumes. However, in order to fully grasp the concepts, an undergraduate level of mathematics and probability theory is generally required from the reader (see Appendices A and B for a quick review of some basic mathematics and probability theory). In addition, the reader is also advised to refer to the notation section (pages 369–372) on set theory, mathematical and probability symbols used in this book.

1.1 Introduction

We consider an experiment or a trial whose result (outcome) is not predictable with certainty. The set of all possible outcomes of an experiment is called the sample space and we denote it by . Any subset of the sample space is known as an event, where an event is a set consisting of possible outcomes of the experiment.

The collection of events can be defined as a subcollection of the set of all subsets of ...