# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PDS] Probability in Finance: Key Ideas – II

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$\sigma$-field on $\Omega$

Just like the collection of Lebesgue-measurable sets $\mathbb{M}$ represented ‘nice’ subsets of $\mathbb{R}$, for a general sample space $\Omega$ (as in probability ‘experiments’), one can also think of collection of ‘nice’ subsets of $\Omega$ in a similar vein.

The analogue of $\mathbb{M}$ on $\mathbb{R}$ is $\mathbb{F}$ on $\Omega$. $\mathbb{F}$ has the same properties as $\mathbb{M}$, i.e. it is closed under ‘complements’ and ‘finite unions’ as:

1. $\Omega \in \mathbb{F}$

2. $A \in \mathbb{F}$ $\Rightarrow A^c \in \mathbb{F}$

3. $A_i \in \mathbb{F}$ $\forall i \ge 1$ $\displaystyle \Rightarrow \bigcup_{i=1}^{\infty}A_i \in \mathbb{F}$

and defines what is called a $\sigma$-field on $\Omega$.

A $\sigma$-field need not comprise all subsets of $\Omega$ – it could be even for a subset of $\Omega$ as long as it satisfies all the above properties.

For example in a ‘die toss’ experiment with sample space $\Omega = \{1, 2, 3, 4, 5, 6\}$, $\mathbb{F_A} = \{\Omega, \varnothing, \{1, 2\}, \{3, 4, 5, 6\}\}$ is an example of a $\sigma$-field generated by subsets $\mathbb{A} = \{\{1, 2\}, \{3, 4, 5, 6\}\}$.

Needless to say, one can come up with other $\sigma$-fields on $\Omega$ which are ‘larger’ than $\mathbb{F_A}$. For example, consider the collection $\{\{1, 2\}, \{3, 4\}, \{5, 6\}\}$ and the $\sigma$-field generated by that subset. Clearly it would be a larger $\sigma$-field than $\mathbb{F_A}$ because it would not only contain the elements contained in $\mathbb{F_A}$ but also some more.

There is a nice result pertaining to $\sigma$-fields which says that for any given collection of subsets $\mathbb{A}$ of $\mathbb{F}$, there exists a smallest $\sigma$-field that contains $\mathbb{A}$. For example, $\mathbb{F_A}$ above describes the smallest $\sigma$-field containing $\mathbb{A} =\{\{1, 2\}, \{3, 4, 5, 6\}\}$.

The smallest $\sigma$-field containing $\mathbb{A}$ is then referred to as the $\sigma$-field generated by $\mathbb{A}$.

Borel Field and Borel Sets

Although $\mathbb{M}$ is a collection of ‘nice’ subsets of $\mathbb{R}$, it turns it is often still too large for our purpose (of measuring probabilities). What is often required is not $\mathbb{M}$, but some nice $\sigma$-field like $\mathbb{M}$, but perhaps smaller than $\mathbb{M}$, that contains all intervals (closed, open, semi-open/semi-closed – all kinds).

As pointed out earlier, $\mathbb{M}$ contains all intervals and also all null sets. If we apply the result mentioned above that given a collection of subsets $\mathbb{A}$ (all intervals), then we know that there must exist a smallest $\sigma$-field containing all intervals. Since all intervals $\mathbb{A}$ are part of $\mathbb{M}$, if that $\sigma$-field exists, it is ‘included’ in $\mathbb{M}$.

Indeed, such a $\sigma$-field exists, which is the smallest $\sigma$-field containing all intervals, called a Borel field $\mathbb{B}$. The elements $B$ of $\mathbb{B}$ are called Borel sets.

For most purposes in probability, the Borel $\sigma$-field $\mathbb{B}$, it turns out, is good enough. So, we may get by defining measures on this ‘smaller’ $\sigma$-field instead of $\mathbb{M}$.

Restricting Lebesge Measure

We have so far defined measures of the kind $m:\mathbb{M} \rightarrow [0, \infty)$, but we know from our intuitive understanding of probability that a probability measure must lie between $[0, 1]$. So the last piece of machinery we need is something that allows us to ‘restrict’ Lebesgue measure $m$ to any Lebesgue-measurable subset $B \in \mathbb{R}$.

Given the measure space $\big(\mathbb{R}, \mathbb{M}, m\big)$, the following construction ‘restricts’ Lebesgue measure to a Lebesgue-measurable subset $B \in \mathbb{R}$:

$\mathbb{M}_B = \{A \cap B: A \in \mathbb{M} \}$

such that $\forall C \in \mathbb{M}_B$

$m_B(C) = m(C)$

The triple $\big(B, \mathbb{M}_B, m_B\big)$ is then a (complete) measure space.

Probability Space

The fact that the above ‘restriction’ of Lebesgue measure results in a measure space now allows us to define probability measure over arbitrary spaces without worrying about if we can ‘restrict’ that measure to $[0, 1]$.

Probability Space: Definition

A probability space is a triple $\big(\Omega, \mathbb{F}, \mathbb{P} \big)$, where $\Omega$ is an arbitrary set (‘sample space’), $\mathbb{F}$ is a $\sigma$-field of subsets of $\Omega$ (i.e. elements of $\mathbb{F}$ are all possible ‘events’), and $\mathbb{P}$ is a measure on $\mathbb{F}$:

$\mathbb{P}(\Omega) = 1$

called probability measure or simply probability.

(Definitions in this post, as in the previous one, are taken from Capinski and Kopp.)

With an abstract measure space, one can always assign the measure to lie between $[0, 1]$ depending on the nature of the experiment.

In the case when $\Omega$ is a Lebesgue-measurable subset of $\mathbb{R}$ where the measures/lengths may indeed be larger than $1$. But the fact that one can ‘restrict’ measures to Lebesgue-measurable subsets of $\mathbb{R}$ affords us a way out. This is done by writing probability for any subset $B \in \mathbb{M}_{\Omega}$ as:

$\mathbb{P}(B) = \displaystyle \frac{1}{m(\Omega)}m(B)$

where $\mathbb{M}_{\Omega} = \{A \cap \Omega: A \in \mathbb{M} \}$

The measure $\mathbb{P}$ as defined above is a restriction of $m:\mathbb{M} \rightarrow \left[0, \infty\right)$ to $\mathbb{P}:\mathbb{M}_{\Omega} \rightarrow [0, 1]$ and we are guaranteed that $\big(\Omega, \mathbb{M}_{\Omega}, \mathbb{P} \big)$ is a measure/probability space.

Note that as defined above, probability measure $\mathbb{P}$ need not have any physical meaning attached to it. But the construction above ensures that it can handle all kinds of events and sample spaces that we may encounter when dealing with arbitrary (and often infinite) sample spaces.

In the following we take a look at the idea of Lebesgue-measurable functions which will take us to the important notion of random variables.

Written by Vineet

February 14, 2013 at 2:24 am

Posted in Teaching: PDS

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