The building blocks

A starting point for our analysis is a probability triple $\left( \Omega ,%% \mathscr{F},P\right)$ . The ingredients of this triple are defined in the following way:

Armed with a probability triple of the above form we can define a random variable

as a mapping from [subsets of] $\Omega$ to $\mathbbm{R}\cup \left\{ \pm \infty \right\}$ . For example the daily return of a stock price is a random variable. It is a black box that takes the events [for example mergers, strikes, floods, etc.] and gives a real number [the stock price return]. Random experiments have qualitative aspects, like mergers above. A random variable gives a quantitative flavor to those events, in order for one to handle them effectively.

The set $\Omega$ in this particular case is massive, since it includes everything that affects the asset price is any possible way. Theoretically speaking, the $\sigma$ -algebra would consist of collections of subsets of $\Omega$ , the possible events that are of interest. One can see why the minimal algebra is equivalent of knowing nothing, since the only events that are considered are the impossible [ $\emptyset$ ] and the certain [ $\Omega$ ] ones. The maximal algebra is equivalent to knowing everything, since the events considered are as detailed as possible [an event such as the hangover of a temporary clerk in the department of agriculture in the bank of Greenland would be included, since it affects the world markets in a chaotic manner]. In general, a typical event $\sigma$ -algebra would lie somewhere between, for instance a rise of the

When the sample space is just the real line, $\Omega =\mathbbm{R}$ , we can define the Borel $\sigma$ -algebra as the smallest $\sigma$ -algebra containing all open intervals in $\mathbbm{R}$ . This would be the case if the stock return itself was considered the random experiment itself, ignoring its underlying causes. In this case, the random variable

would just be the identity function. We denote this algebra by $\mathscr{B}\left( \mathbbm{R}\right)$ , and in a similar fashion we can construct $\mathscr{B}\left( \mathbbm{R}^{k}\right)$ . The properties listed above imply that all closed, semiopen and semiclosed intervals, as well as their unions belong to $\mathscr{B}\left( \mathbbm{R}\right)$ . We will also define the Lebesgue measure as the function that maps each interval to its length. We will denote this function with $\mu _{0}:\mathscr{B}\left( \mathbbm{R}\right) \rightarrow \mathbbm{R}_{+}$ .

Example 34 (Standard normal distribution) We start by considering the space $\left( \mathbbm{R},\mathscr{B}\left( \mathbbm{R}\right) \right)$ . Define the function $\phi \left( x\right) =\left( 2\pi \right) ^{-1/2}\exp \left\{ -x^{2}/2\right\}$ , which is just the standardized normal density. Define the function

as the Lebesgue integral

$\displaystyle P\left( F\right) =\int_{F}\phi d\mu _{0}$ .

If the set $F=\left[ \alpha ,\beta \right]$ we can then rewrite the above in the more familiar form

$\displaystyle P\left( \left[ \alpha ,\beta \right] \right) =\int_{\alpha }^{\beta }\frac{1}{\sqrt{2\pi }}e^{-\frac{x^{2}}{2}}dx$ .

A random variable itself can generate a $\sigma$ -algebra. Suppose that we start with a random variable $Y:\Omega \rightarrow \mathbbm{R}$ . We will say that a set $A\subset \Omega$ is determined by

if by the value of $Y\left( \omega \right) ,\omega \in \Omega$ alone we can determine whether or not $\omega \in A$ . The collection of subsets of $\Omega$ determined by

forms a $\sigma$ -algebra, which is called the $\sigma$ -algebra generated by

and is denoted by $\sigma \left( Y\right)$ . Apparently, just by observing the stock prices one cannot conclude whether the price went down because of a specific strike action (which could be the event $\omega _{1}$ ) or because of a particular interest rates increase (which could be the event $\omega _{2}$ ): these kind of events alone which belong to $\Omega$ cannot be determined by the stock price, and therefore will not be members of $\sigma \left( Y\right)$ , $\left\{ \omega _{1}\right\} \notin \sigma \left( Y\right)$ and $\left\{ \omega _{2}\right\} \notin \sigma \left( Y\right)$ . On the other hand, collections of such events will belong to $\sigma \left( Y\right)$ : if only $\omega _{1}$ or $\omega _{2}$ make the stock return take the observed value, then $\left\{ \omega _{1},\omega _{2}\right\} \in \sigma \left( Y\right)$ . If $\mathscr{F}=\mathscr{B}\left( \mathbbm{R}\right)$ , then in general $\sigma \left( Y\right) =\left\{ Y^{-1}\left( B\right) :B\in \mathscr{B}\left( \mathbbm{R}\right) \right\}$ . One can now see the notion of a filtration which is generated by a random variable: by observing the random variable we understand more and more about what its underlying causes have been.

We will now turn to the definition of the conditional expectation. To illustrate the importance of the $\sigma$ -algebra that we defined above we start with an discrete sample space example.

Now we can turn into defining the conditional expectation of the stock price. Apparently we have to begin by introducing a probability measure [so far we have only dealt with the space $\left( \Omega ,\mathscr{F}\right)$ and the subspaces $\left( \Omega ,\mathscr{F}_{k}\right)$ ]. So let's denote with $p\in \left( 0,1\right)$ the probability of heads, and let us also assume that the tosses are independent. This implies that $P\left( \omega \right) =P\left( \omega _{1},\omega _{2},\omega _{3}\right) =p^{n}\left( 1-p\right) ^{3-n}$ , where

is the number of heads in the set $\left\{ \omega _{1},\omega _{2},\omega _{3}\right\}$ . Then of course for every subset $A\subseteq \Omega$ we define

More formally, let $\left( \Omega ,\mathscr{F},P\right)$ be a probability space. In addition let $\mathscr{G}$ be a sub- $\sigma$ -algebra of $\mathscr{F}$ , and

a random variable on $\left( \Omega ,\mathscr{F},P\right)$ . Then $%% \mathbf{E}\left[ Y\vert\mathscr{G}\right]$ is defined to be a random variable that satisfies:

Such a random variable always exists and it is unique with probability one. For two random variables

and

we also write $\mathbf{E}\left[ Y\vert X\right]$ and we mean $\mathbf{E}\left[ Y\vert\sigma \left( X\right) \right]$ . There are two things to remember:

Example 40 (cont. Stock price and coin tosses) Say that we want to compute $\mathbf{E}\left[ S_{2}\vert\mathscr{F}_{1}\right]$ . Recall that $\mathscr{F}_{1}=\left\{ \emptyset ,\Omega ,F_{H},F_{T}\right\}$ . In addition, $\mathbf{E}\left[ S_{2}\vert\mathscr{F}_{1}%% \right]$ must be constant on $F_{H}$ and $F_{T}$ , since it is $\mathscr{F}%% _{1}$ -measurable. The partial averaging property implies that

$\displaystyle \int_{F_{H}}\mathbf{E}\left[ S_{2}\vert\mathscr{F}_{1}\right] \left( \omega \right) dP\left( \omega \right)$	$\displaystyle =$	$\displaystyle \int_{F_{H}}S_{2}\left( \omega \right) dP\left( \omega \right)$ , and
$\displaystyle \int_{F_{T}}\mathbf{E}\left[ S_{2}\vert\mathscr{F}_{1}\right] \left( \omega \right) dP\left( \omega \right)$	$\displaystyle =$	$\displaystyle \int_{F_{T}}S_{2}\left( \omega \right) dP\left( \omega \right)$ .

We can compute

$\displaystyle \int_{F_{H}}\mathbf{E}\left[ S_{2}\vert\mathscr{F}_{1}\right] \left( \omega \right) dP\left( \omega \right)$	$\displaystyle =$	$\displaystyle P\left( F_{H}\right) \mathbf{E}\left[ S_{2}\vert\mathscr{F}_{1}\right] \left( \omega \right)$
	$\displaystyle =$	$\displaystyle p\mathbf{E}\left[ S_{2}\vert\mathscr{F}_{1}\right] \left( \omega \right) ,\forall \omega \in F_{H}$ .

In addition,

$\displaystyle \int_{F_{H}}S_{2}\left( \omega \right) dP\left( \omega \right) =p^{2}u^{2}S_{0}+p\left( 1-p\right) udS_{0}$ .

The above two equations can be solved to yield

$\displaystyle \mathbf{E}\left[ S_{2}\vert\mathscr{F}_{1}\right] \left( \omega \right)$	$\displaystyle =$	$\displaystyle pu^{2}S_{0}+\left( 1-p\right) udS_{0}$
	$\displaystyle =$	$\displaystyle \left[ pu+\left( 1-p\right) d\right] uS_{0}$
	$\displaystyle =$	$\displaystyle \left[ pu+\left( 1-p\right) d\right] S_{1}\left( \omega \right) ,\forall \omega \in F_{H}$ .

Similarly

$\displaystyle \mathbf{E}\left[ S_{2}\vert\mathscr{F}_{1}\right] \left( \omega \... ...\left( 1-p\right) d\right] S_{1}\left( \omega \right) ,\forall \omega \in F_{T}$ ,

which conclude with the familiar

$\displaystyle \mathbf{E}\left[ S_{2}\vert\mathscr{F}_{1}\right] \left( \omega \... ...left( 1-p\right) d\right] S_{1}\left( \omega \right) ,\forall \omega \in \Omega$ .

$\displaystyle S_{1}\left( \omega \right)$	$\displaystyle =$	$\displaystyle S_{1}\left( \omega _{1},\omega _{2},\omega _{3}\right) =S_{1}\left( \omega _{1}\right)$ ,
$\displaystyle S_{2}\left( \omega \right)$	$\displaystyle =$	$\displaystyle S_{2}\left( \omega _{1},\omega _{2},\omega _{3}\right) =S_{2}\left( \omega _{1},\omega _{2}\right)$ , and
$\displaystyle S_{3}\left( \omega \right)$	$\displaystyle =$	$\displaystyle S_{3}\left( \omega _{1},\omega _{2},\omega _{3}\right)$ .

$\displaystyle \mathbf{E}Y$	$\displaystyle =$	$\displaystyle \sum_{\omega \in \Omega }Y\left( \omega \right) P\left( \omega \right)$ , or
$\displaystyle \mathbf{E}Y$	$\displaystyle =$	$\displaystyle \int_{\Omega }Y\left( \omega \right) dP\left( \omega \right)$ .

$\displaystyle \mathbf{E}\mathtt{I}_{A}Y$	$\displaystyle =$	$\displaystyle \sum_{\omega \in A}Y\left( \omega \right) P\left( \omega \right)$ , or
$\displaystyle \mathbf{E}\mathtt{I}_{A}Y$	$\displaystyle =$	$\displaystyle \int_{A}Y\left( \omega \right) dP\left( \omega \right)$ .

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The building blocks