Measure Theory Q's wrt Stochastic Processes

  • #1

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Hello there.

The stochastic calc book i'm going through ( and others I've seen ) uses the phrase "[itex]\mathscr{F}-measurable[/itex]" random variable [itex]Y[/itex] in the section on measure theory. What does this mean? I'm aware that [itex] \mathscr{F}[/itex] is a [itex]\sigma-field[/itex] over all possible values for the possible values of [itex]Y[/itex], but the language throws me.

A tougher thing that confuses is me is in the stochastic context it defines a filtration as the increasing family of sigma algebras indexed by [itex]t[/itex], like this [itex]\mathscr{F}_t = \sigma{\lbrace X_s; s \leq t \rbrace }[/itex] for a stochastic process [itex]X_t[/itex]. I take it [itex]\sigma[/itex] is an operator here? I realize this is a theoretical construct, but is it possible to see an example (not given in the book). From what I understand [itex] \mathscr{F}_s \subset \mathscr{F}_t \forall s < t[/itex], but again what do either of these look like. Or a simpler, yet nontrivial example of a filtration? Maybe it would be easier in discrete stochastic process case. I can understand how most of this works simply by the fact that no definitions seem conflict with each other and everything is logically consistent but I'm just wondering if there is some more practical ground here ...
 
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  • #2
jbunniii
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Hello there.

The stochastic calc book i'm going through ( and others I've seen ) uses the phrase "[itex]\mathscr{F}-measurable[/itex]" random variable [itex]Y[/itex] in the section on measure theory. What does this mean? I'm aware that [itex] \mathscr{F}[/itex] is a [itex]\sigma-field[/itex] over all possible values for the possible values of [itex]Y[/itex], but the language throws me.
I'm very surprised if they use that terminology without defining it. A measurable function ("random variable" in probability theory) is simply a function ##f## for which ##f^{-1}(A)## is measurable whenever ##A## is measurable.

Your random variable ##Y## is defined on some space, let's call it ##S##, and it is equipped with a sigma algebra ##\mathcal{F}##. You didn't specify the target space, so let's call it ##T## and assume it is also a measure space with some sigma algebra ##\mathcal{G}##. Then ##Y : S \rightarrow T##, and "##Y## is measurable" means that for any set ##A \in \mathcal{G}##, we have ##f^{-1}(A) \in \mathcal{F}##.
 
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  • #3
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For an example, let's have a probability space [itex](\Omega, \mathcal{F}, \mathbb{P})[/itex] and a stochastic process (coin tossing) [itex]X[/itex] adapted to a filtration [itex](\mathcal{F}_t)_{t=1,2}, \mathcal{F}_t\subset \mathcal{F}[/itex].

Then, in the beginning, one knows nothing but possible results,
[itex]\mathcal{F}_0 = \{\emptyset, \Omega\}[/itex] where [itex]\Omega=\{[HH], [HT], [TH], [TT]\}[/itex].

After the first toss, the division becomes finer, one can say more about the history,
[itex]\mathcal{F}_1 = \{\emptyset, \Omega, [HH, HT], [TH, TT]\} \supset \mathcal{F}_0[/itex],

After the second toss,
[itex]\mathcal{F}_2 = \{\emptyset, \Omega, [HH, HT], [TH, TT], [HH], [HT], [TH], [TT]\} \supset{F}_1[/itex].

You can exchange 2 tosses of a coin by [itex]m[/itex] tosses of a dice, [itex]\Omega=\{1,...,6\}^m[/itex], or even to move to continuous cases.
 
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  • #4
Both of your answers were incredibly helpful. Thank you.
 

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