Why is the measurability condition in the definition of a predictable process the same as "the value at $t$ is know at $t-1$"?

Asked Apr 21 '17 at 03:19

Active Apr 21 '17 at 03:28

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First of all, here is a related question asked: What exactly is a 'predictable process'?

My question is, why $X_t$ is measurable with respect to $\mathcal{F}_{t-1}$ the same as the value of $X_t$ is known at $t-1$ ?

A similar question can be asked about an adapted process.

Let's focus on discrete processes for the moment.

Thank you very much.

edited Apr 21 '17 at 03:28

asked Apr 21 '17 at 03:19

Spartan 117

4

You don't need stochastic processes to see this, it is a more basic theorem that if $Y $ is $\sigma (X) $ measurable then there is a Borel function $f $ such that $Y=f(X) $. This should be in any measure theoretic probability book, though not under any particular name. – Ian Apr 21 '17 at 03:33
@lan This answers the question! – Spartan 117 Apr 21 '17 at 07:40

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