Chapter 1
General 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 and we define any collection of subsets of as a field if it satisfies the following.

Definition 1.1

The sample space is the set of all possible outcomes of an experiment or random trial. A field is a collection (or family) of subsets of with the following conditions:

1. where is the empty set;
2. if then where is the complement of in ;
3. if , then —that is to say, is closed under finite unions.

It should be noted in the definition of a field that is closed under finite unions (as well as under finite intersections). As for the case of a collection of events closed under countable unions (as well as under countable intersections), any collection of subsets of with such properties is called a -algebra.

Definition 1.2

If is a given sample space, then a -algebra (or -field) on is a family (or collection) of subsets of with the following properties:

1. ;
2. if then where is the complement of in ;
3. if then —that is to say, is closed under countable unions.

We next outline an approach to probability which is a branch of measure theory. The reason for taking a measure-theoretic path is that it leads to a unified treatment of both discrete and continuous random variables, as well as a general definition of conditional expectation.

Definition 1.3

The pair is called a measurable space. A probability measure on a measurable space is a function such that:

1. ;
2. ;
3. if and is disjoint such that , then .

The triple is called a probability space. It is called a complete probability space if also contains subsets of with -outer measure zero, that is .

By treating -algebras as a record of information, we have the following definition of a filtration.

Definition 1.4

Let be a non-empty sample space and let be a fixed positive number, and assume for each there is a -algebra . In addition, we assume that if , then every set in is also in . We call the collection of -algebras , , a filtration.

Below we look into the definition of a real-valued random variable, which is a function that maps a probability space to a measurable space .

Definition 1.5

Let be a non-empty sample space and let be a -algebra of subsets of . A real-valued random variable is a function such that for each and we say is measurable.

In the study of stochastic processes, an adapted stochastic process is one that cannot “see into the future” and in mathematical finance we assume that asset prices and portfolio positions taken at time are all adapted to a filtration , which we regard as the flow of information up to time . Therefore, these values must be measurable (i.e., depend only on information available to investors at time ). The following is the precise definition of an adapted stochastic process.

Definition 1.6

Let be a non-empty sample space with a filtration , and let be a collection of random variables indexed by . We therefore say that this collection of random variables is an adapted stochastic process if, for each , the random variable is measurable.

Finally, we consider the concept of conditional expectation, which is extremely important in probability theory and also for its wide application in mathematical finance such as pricing options and other derivative products. Conceptually, we consider a random variable defined on the probability space and a sub--algebra of (i.e., sets in are also in ). Here can represent a quantity we want to estimate, say the price of a stock in the future, while contains limited information about such as the stock price up to and including the current time. Thus, constitutes the best estimation we can make about given the limited knowledge . The following is a formal definition of a conditional expectation.

Definition 1.7 (Conditional Expectation)

Let be a probability space and let be a sub--algebra of (i.e., sets in are also in ). Let be an integrable (i.e., ) and non-negative random variable. Then the conditional expectation of given , denoted , is any random variable that satisfies:

1. is measurable;
2. for every set , we have the partial averaging property

From the above definition, we can list the following properties of conditional expectation. Here is a probability space, is a sub--algebra of and is an integrable random variable.

• Conditional probability. If is an indicator random variable for an event then
• Linearity. If , , , are integrable random variables and , , , are constants then
• Positivity. If almost surely then almost surely.
• Monotonicity. If and are integrable random variables and almost surely then
• Computing expectations by conditioning. .
• Taking out what is known. If and are integrable random variables and is measurable then
• Tower property. If is a sub--algebra of then
• Measurability. If is measurable then .
• Independence. If is independent of then .
• Conditional Jensen's inequality. If is a convex function then

1.2 Problems and Solutions

1.2.1 Probability Spaces

1. 1. De Morgan's Law. Let , where is some, possibly uncountable, indexing set. Show that
   1. .
   2. .

   Solution

   1. Let which implies , so that for all . Therefore,

      On the contrary, if we let then for all or and hence

      Therefore, .

   2. From (a), we can write

      Taking complements on both sides gives

2. 2. Let be a -algebra of subsets of the sample space . Show that if then .

   Solution

   Given that is a -algebra then and . Furthermore, the complement of is .

   Thus, from De Morgan's law (see Problem 1.2.1.1, page 4) we have .

3. 3. Show that if is a -algebra of subsets of then .

   Solution

   is a -algebra of subsets of , hence if then .

   Since then . Thus, .

4. 4. Show that if then is a -algebra of subsets of .

   Solution

   is a -algebra of subsets of since

   1. .
   2. For then . For then . In addition, for then . Finally, for then .
   3. , , , , , , and .
5. 5. Let , be a family of -algebras of subsets of the sample space . Show that is also a -algebra of subsets of .

   Solution

   is a -algebra by taking note that

   1. Since , therefore as well.
   2. If for all then , . Therefore, and hence .
   3. If for all then , and hence and .

   From the results of (a)–(c) we have shown is a -algebra of .

6. 6. Let and let

   Show that and are -algebras of subsets of .

   Is also a -algebra of subsets of ?

   Solution

   Following the steps given in Problem 1.2.1.4 (page 5) we can easily show and are -algebras of subsets of .

   By setting , and since and , but , then is not a -algebra of subsets of .

7. 7. Let be a -algebra of subsets of and suppose so that . Show that .

   Solution

   Given that and are mutually exclusive we therefore have

   Thus, we can express

   Since and therefore .

8. 8. Let be a probability space and let be defined by where such that . Show that is also a probability space.

   Solution

   To show that is a probability space we note that

   1. .
   2. .
   3. Let be disjoint members of and hence we can imply , are also disjoint members of . Therefore,

      Based on the results of (a)–(c), we have shown that is also a probability space.

9. 9. Boole's Inequality. Suppose is a countable collection of events. Show that

   Solution

   Without loss of generality we assume that and define , , such that are pairwise disjoint and

   Because , , we have

10. 10. Bonferroni's Inequality. Suppose is a countable collection of events....