# F-measurable

• Jan 11th 2011, 07:07 PM
Volga
F-measurable
I am a newby in stochastic processes, which I study in the context of the modelling of security prices (financial mathematics). I'd appreciate your help.

Question. Consider the sample space $\displaystyle \Omega=\{-3,-2,-1,1,2,3\}$ and the algebra F=$\displaystyle \{\phi,\{-3,-2\}, \{-1,-1\}, \{2,3\}, \{-3,-2,-1,1\},\{-3,-2,2,3\}, \{-1,1,2,3\}, \Omega\}$.

For each of the following random variables, determine whether it is F-measurable:
(i) $\displaystyle X(\omega)=\omega^2$
(ii) $\displaystyle X(\omega)=max(\omega,2)$.

Find a random variable that is F-measurable.

I look back at the definition of F-measurable: "the random variable X is said to be F-measurable with respect to the algebra F if the function $\displaystyle \omega\rightarrow{X(\omega)}$ is constant on any subset in the partition corresponding to F (Pliska, Introduction to Mathematical Finance).

Therefore I need to check whether

(i) $\displaystyle X(\omega)=\omega^2$ is true for every element of every subset above. Obviously, it is only good for the subset {-1,1}; in any other subset, each individual $\displaystyle \omega$ is not equal to itself squared, so $\displaystyle X(\omega)$ is not the same on each subset excpet for {-1,1}.

(ii)similarly, I apply this function (max, 2) to each component of the subsets listed above, and most of them fail: even if $\displaystyle X(\omega)=max(\omega,2)$ for {-3,-2} and {-1,1}, in {2,3} I have $\displaystyle X(2)=2$$\displaystyle X(3)=3$ and X(2) does not equal to X(3).

So, neither (i) nor (ii) are F-measurable.

To find an F-measurable variable, I borrow idea from (ii):
$\displaystyle X(\omega)=max(\omega,3)$. I think it is F-measurable...

PS Is there 'curly F' in Latex code?...
• Jan 12th 2011, 06:42 AM
Guy
Hello,

That seems like an odd definition of measurability to me; are you sure that's right? Usually one would require that the inverse image of open (or measurable) sets be measurable sets, and it suffices to check that $\displaystyle X^{-1} ([a, \infty))$ is measurable for all $\displaystyle a \in \mathbb{R}$ assuming the Borel algebra on $\displaystyle \mathbb{R}$ (see Real and Complex Analysis by Rudin, for example).

Anyways, I don't think either is measurable because you don't get measurable sets after taking the inverse image of $\displaystyle \{9\}$ and $\displaystyle \{3\}$ respectively. A measurable map is, for example, one that takes everyone to $\displaystyle 0$.

To get the kind of F you want, double click on this: $\displaystyle \mathcal{F}$.
• Jan 13th 2011, 12:57 AM
Volga
No, I am not sure if it's right (or wrong), I am simply restating it from the book. The asset pricing models that use this definition would have asset prices as random variables, several states of nature (with different probabilities) and several periods where information gradually becomes available to investors. For a more precise picture, you can check the following lecture (page 12 for definitiion of $\displaystyle \mathcal{F}$-measurable):

http://www.math.ust.hk/~maykwok/cour...all/Topic2.pdf

Actually, the original question is from the same teaching notes (but I am not taking this course with the lecturer so I cannot ask him!). I attemted the question hoping practice would make me understand the theory better; but right now I am even more confused by your reply )))

I checked Grimmet "Probability and Random Processes" and he gives the following definition:

A random variable is a function $\displaystyle X: \Omega\rightarrowR$ with the property that {$\displaystyle \omega\in\Omega: X(\omega)\legx$}$\displaystyle \in\mathcal{F}$ for each $\displaystyle x\inR$. Such function is said to be $\displaystyle \mathcal{F}$-measurable.

I am still hoping to get a bit closer to understanding $\displaystyle \mathcal{F}$-measurable with the help of this Forum. Thank you for your time!
• Jan 13th 2011, 07:46 AM
Guy
It's hard to know what to say without having the material or anything (the notes given didn't have the definition). Wikipedia gives the usual definition of measurable functions, the issue with your question being that you haven't specified the measurable sets in the range of the functions (usually it can be inferred that it's the Borel subsets of the real numbers with the usual topology).

I don't know much mathematical finance so I can't really help you beyond that. I think you made a mistake typing up the definition from Grimmet, but it looks to me like it's probably a restricted version of the more general definition given above.
• Jan 13th 2011, 10:49 PM
Volga
Guy, I appreciate your time spent. I decided to read up more on probability first before going back to the Finance book. I may be able to get back here with an answer one day )))