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The technical definition of an adapted stochastic process can be found here https://en.wikipedia.org/wiki/Adapted_process.

I understand the following chain of consequences from this definition:

[itex]{X_i}[/itex] is adapted

[itex]\Rightarrow[/itex] Each random variable [itex]X_i[/itex] is measurable with respect to the filtration [itex]\mathcal{F}_i[/itex]

[itex]\Rightarrow[/itex] The preimage of any Borel set under the map [itex]X_i[/itex] is in the filtration [itex]\mathcal{F}_i[/itex]

[itex]\Rightarrow[/itex] It is possible to define the probability [itex]P(X_i \in B)[/itex] for all Borel sets [itex]B[/itex].

What I don't understand is the following line in the Wikipedia article "An informal interpretation is that [itex]{X_i}[/itex] is adapted if and only if, for every realization and every [itex]i[/itex], [itex]X_i[/itex] is known at time [itex]i[/itex]".

How does this follow from the definition?

It seems to me that "measurable with respect to the filtration [itex]\mathcal{F}_i[/itex]" means we can put a probability on [itex]X_i[/itex] being in some set of values, [itex]B[/itex], at time [itex]i[/itex], but the above assertion seems to go one step further, that we can know the value of [itex]X_i[/itex] with certainty at time [itex]i[/itex]. Why does an adapted process have this interpretation?

I understand the following chain of consequences from this definition:

[itex]{X_i}[/itex] is adapted

[itex]\Rightarrow[/itex] Each random variable [itex]X_i[/itex] is measurable with respect to the filtration [itex]\mathcal{F}_i[/itex]

[itex]\Rightarrow[/itex] The preimage of any Borel set under the map [itex]X_i[/itex] is in the filtration [itex]\mathcal{F}_i[/itex]

[itex]\Rightarrow[/itex] It is possible to define the probability [itex]P(X_i \in B)[/itex] for all Borel sets [itex]B[/itex].

What I don't understand is the following line in the Wikipedia article "An informal interpretation is that [itex]{X_i}[/itex] is adapted if and only if, for every realization and every [itex]i[/itex], [itex]X_i[/itex] is known at time [itex]i[/itex]".

How does this follow from the definition?

It seems to me that "measurable with respect to the filtration [itex]\mathcal{F}_i[/itex]" means we can put a probability on [itex]X_i[/itex] being in some set of values, [itex]B[/itex], at time [itex]i[/itex], but the above assertion seems to go one step further, that we can know the value of [itex]X_i[/itex] with certainty at time [itex]i[/itex]. Why does an adapted process have this interpretation?

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