- #1

X89codered89X

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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 ...

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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