0
$\begingroup$

My question is concerned with some notation I came across and can't seem to figure out. Let me give you the background:

Let $(\Omega, \mathcal{F}, P)$ be our probability space, $X, Y, Z : \Omega \rightarrow \mathbb{R}$ random variables, where $X$ is integrable. Then, we define the conditional expectation of $X$ on $Y, Z$, written $\mathbb{E}(X \, | \, Y, Z)$, as the random variable (defined on $\Omega$ with codomain $\mathbb{R}$) such that

  • $\mathbb{E}(X \, | \, Y, Z)$ is $\sigma(Y, Z)$ - measurable;

  • For every $A \in \sigma(Y, Z)$,

\begin{equation*} \int_A \mathbb{E}(X \, | \, Y, Z) \; dP = \int_A X \; dP. \end{equation*}

Notice that the Doob - Dynkin factorization lemma implies that, we can find a function $f : \mathbb{R} \rightarrow \mathbb{R}^2$ such that $\mathbb{E}(X \, | \, Y, Z) = f \circ (Y, Z)$, where $(Y, Z)$ is the random vector with coordinates $Y$ and $Z$. It is standard to write $\mathbb{E}(X \, | \, Y = y, Z = z)$ for the function $f$.

With these definitions in mind my question is: what does the notation $\mathbb{E}(X \, | \, Y, Z = z)$ mean?

As a follow up: Does anyone have a good reference for these concepts? I have good measure theory books, but none dive into the concept I'm asking.

$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

I think you mean that $\mathbb{E}\left(X \, \middle|\, Y,Z\right)$ is $\sigma(Y,Z)$ - measurable, not $\sigma(X,Y)$ - measurable.

The notation $\mathbb{E}\left(X \, \middle|\, Y,Z=z\right)$ or $f(Y,z)$ is just a random variable which depends on $z$ and is a function of $Y$ (i.e. $\sigma(Y)$ - measurable).

For further reference, have a look at any of

  • Borovkov, Probability Theory
  • Karr, Probability
  • Shiryaev, Probability

Measure theory books may discuss this stuff but usually not in as much details, seeing as conditional expectation is more of an application of a measure theoretic concept (the Lebesgue-Radon-Nikodym derivative).

$\endgroup$
3
  • $\begingroup$ Corrected the mistake, thank you. With regard to the definition, If I understood you correctly, the notation $\mathbb{E}(X \, | \, Y, Z = z)$ stands for a function defined in the following way: $\mathbb{E}(X \, | \, Y, Z = z) : \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ such that $\mathbb{E}(X \, | \, Y, Z = z) (\omega) = f(Y(\omega), z)$, right? Also, I look at the references you provided but none of them define this concept explicitly. Do you have any other books which discuss these concepts more thoroughly? $\endgroup$
    – Giordano Ribeiro
    Commented Jun 5, 2023 at 2:21
  • $\begingroup$ The definition looks good. $\endgroup$
    – David Dinh
    Commented Jun 5, 2023 at 3:10
  • 1
    $\begingroup$ On page 91 of Probability Theory, this notion of substitution was discussed. If you are looking for a very specific definition of the above, I am afraid you will not find it here but I think the discussion will suffice. I doubt you will find a very specific definition of this concept. Usually writers don't include these kind of definitions, they take them for granted and assume that readers can figure out the details (which you seem to have). $\endgroup$
    – David Dinh
    Commented Jun 5, 2023 at 3:30

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .