Stochastic Processes and Financial Mathematics
(part one)

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Chapter 2 Probability spaces and random variables

In this chapter we review probability theory, and develop some key tools for use in later chapters. We begin with a special focus on $\sigma $-fields. The role of a $\sigma $-field is to provide a way of controlling which information is visible (or, currently of interest) to us. As such, $\sigma $-fields will allow us to express the idea that, as time passes, we gain information.

2.1 Probability measures and $\sigma $-fields

Let $\Omega $ be a set. In probability theory, the symbol $\Omega $ is typically (and always, in this course) used to denote the sample space. Intuitively, we think of ourselves as conducting some random experiment, with an unknown outcome. The set $\Omega $ contains an $\omega \in \Omega $ for every possible outcome of the experiment. It is alternatively known as the state space.

Subsets of $\Omega $ correspond to collections of possible outcomes; such a subset is referred as an event. For instance, if we roll a dice we might take $\Omega =\{1,2,3,4,5,6\}$ and the set $\{1,3,5\}$ is the event that our dice roll is an odd number.

Definition 2.1.1 Let $\mc {F}$ be a set of subsets of $\Omega $. We say $\mc {F}$ is a $\sigma $-field if it satisfies the following properties:
- 1. $\emptyset \in \mc {F}$ and $\Omega \in \mc {F}$.
- 2. if $A\in \mc {F}$ then $\Omega \sc A\in \mc {F}$.
- 3. if $A_1,A_2,\ldots \in \mc {F}$ then $\bigcup _{i=1}^\infty A_i\in \mc {F}$.

The role of a $\sigma $-field is to choose which subsets of outcomes we are actually interested in. The power set $\mc {F}=\mc {P}(\Omega )$ is always a $\sigma $-field, and in this case every subset of $\Omega $ is an event. But $\mc {P}(\Omega )$ can be very big, and if our experiment is complicated, with many or even infinitely many possible outcomes, we might want to consider a smaller choice of $\mc {F}$ instead.

Sometimes we will need to deal with more than one $\sigma $-field at a time. A $\sigma $-field $\mc {G}$ such that $\mc {G}\sw \mc {F}$ is known as a sub-$\sigma $-field of $\mc {F}$.

We say that a subset $A\sw \Omega $ is measurable, or that it is an event (or measurable event), if $A\in \mc {F}$. To make to it clear which $\sigma $-field we mean to use in this definition, we sometimes write that an event is $\mc {F}$-measurable.

Example 2.1.2 Some examples of experiments and the $\sigma $-fields we might choose for them are the following:
- • We toss a coin, which might result in heads $H$ or tails $T$. We take $\Omega =\{H,T\}$ and $\mc {F}=\big \{\emptyset ,\{H\},\{T\},\Omega \big \}$.
- • We toss two coins, both of which might result in heads $H$ or tails $T$. We take $\Omega =\{HH,TT,HT,TH\}$. However, we are only interested in the outcome that both coins are heads. We take $\mc {F}=\big \{\emptyset ,\{HH\},\Omega \sc \{HH\},\Omega \big \}$.

There are natural ways to choose a $\sigma $-field, even if we think of $\Omega $ as just an arbitrary set. For example, $\mc {F}=\{\Omega ,\emptyset \}$ is a $\sigma $-field. If $A$ is a subset of $\Omega $, then $\mc {F}=\{\Omega ,A,\Omega \sc A,\emptyset \}$ is a $\sigma $-field (check it!).

Given $\Omega $ and $\mc {F}$, the final ingredient of a probability space is a measure $\P $, which tells us how likely the events in $\mc {F}$ are to occur.

Definition 2.1.3 A probability measure $\P $ is a function $\P :\F \to [0,1]$ satisfying:
- 1. $\P [\Omega ]=1$.
- 2. If $A_1,A_2,\ldots \in \F $ are pair-wise disjoint (i.e. $A_i\cap A_j=\emptyset $ for all $i,j$ such that $i\ne j$) then
  
  \[ \P \left [ \bigcup _{i=1}^\infty A_i \right ] = \sum _{i=1}^\infty \P [A_i]. \]

The second of these conditions if often called $\sigma $-additivity. Note that we needed Definition 2.1.1 to make sense of Definion 2.1.3, because we needed something to tell us that $\P \left [ \bigcup _{i=1}^\infty A_i \right ]$ was defined!

Definition 2.1.4 A probability space is a triple $(\Omega ,\F ,\P )$, where $\mc {F}$ is a $\sigma $-field and $\P $ is a probability measure.

For example, to model a single fair coin toss we would take $\Omega =\{H,T\}$, $\mc {F}=\{\Omega ,\{H\},\{T\},\emptyset \}$ and define $\P [H]=\P [T]=\frac 12$.

We commented above that often we want to choose $\mc {F}$ to be smaller than $\mc {P}(\Omega )$, but we have not yet shown how to choose a suitably small $\mc {F}$. Fortunately, there is a general way of doing so, for which we need the following technical lemma.

Lemma 2.1.5 Let $I$ be any set and for each $i\in I$ let $\mc {F}_i$ be a $\sigma $-field. Then
$\seteqnumber{0}{2.}{0}$
\begin{equation} \label {eq:sigma_intersect_eq} \mc {F}=\bigcap _{i\in I}\mc {F}_i \end{equation}

is a $\sigma $-field

Proof: We check the three conditions of Definition 2.1.1 for $\mc {F}$.

(1) Since each $\mc {F}_i$ is a $\sigma $-field, we have $\emptyset \in \mc {F}_i$. Hence $\emptyset \in \cap _i \mc {F}_i$. Similarly, $\Omega \in \mc {F}$.

(2) If $A\in \mc {F}=\cap _i \mc {F}_i$ then $A\in \mc {F}_i$ for each $i$. Since each $\mc {F}_i$ is a $\sigma $-field, $\Omega \sc A\in \mc {F}_i$ for each $i$. Hence $\Omega \sc A\in \cap _i \mc {F}_i$.

(3) If $A_j\in \mc {F}$ for all $j$, then $A_j\in \mc {F}_i$ for all $i$ and $j$. Since each $\mc {F}_i$ is a $\sigma $-field, $\cup _j A_j\in \mc {F}_i$ for all $i$. Hence $\cup _j A_j\in \cap _i \mc {F}_i$. ∎

Corollary 2.1.6 In particular, if $\mc {F}_1$ and $\mc {F}_2$ are $\sigma $-fields, so is $\mc {F}_1\cap \mc {F}_2$.

Now, suppose that we have our $\Omega $ and we have a finite or countable collection of $E_1,E_2,\ldots \sw \Omega $, which we want to be events. Let $\mathscr {F}$ be the set of all $\sigma $-fields that contain $E_1,E_2,\ldots $. We enumerate $\mathscr {F}$ as $\mathscr {F}=\{\mc {F}_i\-i\in I\}$, and apply Lemma 2.1.5. We thus obtain a $\sigma $-field $\mc {F}$, which contains all the events that we wanted.

The key point here is that $\mc {F}$ is the smallest $\sigma $-field that has $E_1,E_2,\ldots $ as events. To see why, note that by (2.1), $\mc {F}$ is contained inside any $\sigma $-field $\mc {F}'$ which has $E_1,E_2,\ldots $ as events.

Definition 2.1.7 Let $E_1,E_2,\ldots $ be subsets of $\Omega $. We write $\sigma (E_1,E_2,\ldots ,)$ for the smallest $\sigma $-field containing $E_1,E_2,\ldots $.

With $\Omega $ as any set, and $A\sw \Omega $, our example $\{\emptyset ,A,\Omega \sc A,\Omega \}$ is clearly $\sigma (A)$. In general, though, the point of Definition 2.1.7 is that we know useful $\sigma $-fields exist without having to construct them explicitly.

In the same style, if $\mc {F}_1,\mc {F}_2\ldots $ are $\sigma $-fields then we write $\sigma (\mc {F}_1,\mc {F}_2,\ldots )$ for the smallest $\sigma $-algebra with respect to which all events in $\mc {F}_1, \mc {F}_2, \ldots $ are measurable.

From Definition 2.1.1 and 2.1.3 we can deduce all the ‘usual’ properties of probability. For example:

• If $A\in \mc {F}$ then $\Omega \sc A\in \mc {F}$, and since $\Omega =A\cup (\Omega \sc A)$ we have $\P [\Omega ]=1=\P [A]+\P [\Omega \sc A].$
• If $A,B\in \mc {F}$ and $A\sw B$ then we can write $B=A\cup (B\sc A)$, which gives us that $\P [B]=\P [B\sc A]+P[A]$, which implies that $\P [A]\leq \P [B]$.

And so on. In this course we are concerned with applying probability theory rather than with relating its properties right back to the definition of a probability space; but you should realize that it is always possible to do so.

Definitions 2.1.1 and 2.1.3 both involve countable unions. It is convenient to be able to use countable intersections too, for which we need the following lemma.

Lemma 2.1.8 Let $A_1,A_2,\ldots \in \mc {F}$, where $\mc {F}$ is a $\sigma $-field. Then $\bigcap _{i=1}^\infty A_i\in \mc {F}$.

Proof: We can write

\[\bigcap _{i=1}^\infty A_i=\bigcap _{i=1}^\infty \Omega \sc (\Omega \sc A_i)=\Omega \sc \left (\bigcup _{i=1}^\infty \Omega \sc A_i\right ).\]

Since $\mc {F}$ is a $\sigma $-field, $\Omega \sc A_i\in \mc {F}$ for all $i$. Hence also $\bigcup _{i=1}^\infty \Omega \sc A_i\in \mc {F}$, which in turn means that $\Omega \sc (\bigcup _{i=1}^\infty \Omega \sc A_i)\in \mc {F}$. ∎

In general, uncountable unions and intersections of measurable sets need not be measurable. The reasons why we only allow countable unions/intersections in probability are complicated and beyond the scope of this course. Loosely speaking, the bigger we make $\mc {F}$, the harder it is to make a probability measure $\P $, because we need to define $\P [A]$ for all $A\in \mc {F}$ in a way that satisfies Definition 2.1.3. Allowing uncountable set operations would (in natural situations) result in $\mc {F}$ being so large that it would be impossible to find a suitable $\P $.

From now on, the symbols $\Omega $, $\mc {F}$ and $\P $ always denote the three elements of the probability space $(\Omega ,\mc {F},\P )$.

Stochastic Processes and Financial Mathematics (part one)

Chapter 2 Probability spaces and random variables

2.1 Probability measures and \(\sigma \)-fields

Stochastic Processes and Financial Mathematics
(part one)