Stochastic Processes and Financial Mathematics
(part one)

2.4 Expectation

There is only one part of the ‘usual machinery’ for probability that we haven’t yet discussed, namely expectation.

Recall that the expectation of a discrete random variable \(X\) that takes the values \(\{x_i:i\in \N \}\) is given by

\begin{equation} \label {eq:E_discrete} \E [X]=\sum _{x_i} x_i \P [X=x_i]. \end{equation}

For a continuous random variables, the expectation uses an integral against the probability density function,

\begin{equation} \label {eq:E_cont} \E [X]=\int _{-\infty }^\infty x\,f_X(x)\,dx. \end{equation}

Recall also that it is possible for limits (i.e. infinite sums) and integrals to be infinite, or not exist at all.

We are now conscious of the general definition of a random variable \(X\), as an \(\mc {F}\)-measurable function from \(\Omega \) to \(\R \). There are many random variables that are neither discrete nor continuous, and for such cases (2.4) and (2.5) are not valid; we need a more general approach.

With Lebesgue integration, the expectation \(\E \) can be defined using a single definition that works for both discrete and continuous (and other more exotic) random variables. This definition relies heavily on analysis and is well beyond the scope of this course. Instead, Lebesgue integration is covered in MAS31002/61022.

For purposes of this course, what you should know is: \(\E [X]\) is defined for all \(X\) such that either

• 1. \(X\geq 0\), in which case it is possible that \(\E [X]=\infty \),

• 2. general \(X\) for which \(\E [|X|]<\infty \).

The point here is that we are prepared to allow ourselves to write \(\E [X]=\infty \) (e.g. when the sum or integral in (2.4) or (2.5) tends to \(\infty \)) provided that \(X\geq 0\). We are not prepared to allow expectations to equal \(-\infty \), because we have to avoid nonsensical ‘\(\infty -\infty \)’ situations.

It’s worth knowing that if \(X\geq 0\) and \(\P [X=\infty ]>0\), then \(\E [X]=\infty \). In words, the slightest chance of \(X\) being infinite will outweigh all of the finite possibilities and make \(\E [X]\) infinite.

You may still use (2.4) and (2.5), in the discrete/continuous cases. You may also assume that all the ‘standard’ properties of \(\E \) hold:

You should become familiar with any of the properties that you are not already used to using. The proofs of these properties are part of the formal construction of \(\E \) and are not part of our course.

Indicator functions

One important type of random variable is an indicator function. Let \(A\in \mc {F}\), then the indicator function of \(A\) is the function

\[\1_A(\omega )= \begin {cases} 1 & \omega \in A\\ 0 & \omega \notin A. \end {cases} \]

The indicator function is used to tell if an event occurred (in which case it is \(1\)) or did not occur (in which case it is \(0\)). It is useful to remember that

\[\P [A]=\E [\1_A].\]

We will sometimes not put the \(A\) as a subscript and write e.g. \(\1\{X<0\}\) for the indicator function of the event that \(X<0\).

As usual, let \(\mc {G}\) denote a sub \(\sigma \)-field of \(\mc {F}\).

Proof: Let us write \(Y=\1_A\). Note that \(Y\) is a discrete random variable, which can take the two values \(0\) and \(1\). We have \(Y^{-1}(1)=\{Y=1\}=A\) and \(Y^{-1}(0)=\{Y=0\}=\Omega \sc A\). By Proposition 2.2.2, \(Y\) is \(\mc {G}\) measurable.   ∎

Indicator functions allow us to condition, meaning that we can break up a random variable into two or more cases. For example, given any random variable \(X\) we can write

\begin{equation} \label {eq:condition_ex} X=X\1_{\{X\geq 1\}}+X\1_{\{X<1\}}. \end{equation}

Precisely one of the two terms on the right hand side is non-zero. If \(X\geq 1\) then the first term takes the value \(X\) and the second is zero; if \(X<1\) then the second term is equal to \(X\) and the first term is zero.

We can use (2.6) to prove a useful inequality. Putting \(|X|\) in place of \(X\), and then taking \(\E \) we obtain

\begin{align} \E [|X|]&=\E [|X|\1_{\{|X|\geq 1\}}]+\E [|X|\1_{\{|X|<1\}}]\notag \\ &\leq \E [X^2\1_{\{|X|\geq 1\}}]+1\notag \\ &\leq \E [X^2]+1.\label {eq:L1L2} \end{align} Here, to deduce the second line, the key point is we can only use the inequality \(|x|\leq x^2\) if \(x\geq 1\). Hence, for the first term we can use that \(|X|\1_{\{|X|\geq 1\}}\leq X^2\1_{\{|X|\geq 1\}}\). For the second term, we use that \(|X|\1_{\{|X|<1\}}\leq 1\). In both cases, we also need the monotonicity of \(\E \).

\(L^p\) spaces

It will often be important for to us know whether a random variable \(X\) has finite mean and variance. Some random variables do not, see exercise 2.8 (or MAS2010 or MAS31002) for example. Random variables with finite mean and variances are easier to work with than those which don’t, and many of the results in this course require these conditions.

We use some notation:

In this course, we will only be interested in the cases \(p=1\) and \(p=2\). These cases have the following set of useful properties:

• 1. By definition, \(L^1\) is the set of random variables for which \(\E [|X|]\) is finite.

• 2. \(L^2\) is the set of random variables with finite variance. This comes from the fact that \(\var (X)=\E [X^2]-\E [X]^2\) (more strictly, it needs some integration theory from MAS31002/61022).

• 3. From (2.7), if \(X\in L^2\) then also \(X\in L^1\).

Often, to check if \(X\in L^p\) we must calculate \(\E [|X|^p]\). A special case where it is automatic is the following.

If \(X\) is bounded, then using monotonicity we have \(\E [|X|^p]\leq \E [c^p]=c^p<\infty \), which means that \(X\in L^p\), for all \(p\).