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logarithmic
logarithmic is offline
#5
Mar2-12, 11:59 PM
P: 108
Quote Quote by kai_sikorski View Post
If you know that the realization is in A, then all you don't know is if step 5 was -1, or 1. You know everything else.



Yes you can add this set to F4, but that doesn't mean that the stochastic process will go right 5 times. It means that, you're now allowed to ask whether it did, at time 4.

However adding sets like this to F4, while allowed and would still mean X was adapted is not useful, this is not the natural filtration. The natural filtration is generated by only the information you need at an individual time step to determine the value of the stochastic process. In fact you could make the 5 successive sigma-fields in the filtration F, F, F, F, F, where F is the σ-field for the whole probability space. Again X would be adapted to this filtration, but this would not be useful.

To understand this it really helps to understand the formal measure theoretic interpretation of conditional expectation. See if you can read the wikipedia article on this and understand why [itex]\operatorname{E}(X_i|\mathcal{F}_i):\Omega \to \mathbb{R}[/itex] is not a random variable but [itex]\operatorname{E}(X_i|\mathcal{F}_{i-1}):\Omega \to \mathbb{R}[/itex] is a random variable, although it has much less uncertainty than [itex]X_i[/itex].
Thanks for your reply.

It seems that the misunderstanding is between the math and giving it some real-world interpretation.

Why can't your argument be reversed, i.e:
Yes you have the set A in F4, but that doesn't mean that the stochastic process went -1, 1, 1, 1. It means that, you're now allowed to ask whether it did, at time 4.

I suspect that your answer might be that by time 4 we can obviously observe that the process did go -1, 1, 1, 1 and not 1, 1, 1, 1. But how is that reflected in the math? I think while a natural filtration models the flow of information, not all filtration do?

Are there any nonadapted stochastic processes (that aren't completely pathological)? It seems obvious that we can always know the value of X_t at time t, even if we define a process on t = {1,...,10}, where X_t = X_10 for all t.

It seems I'll have to go away and think about this for a while, particularly the definition on the conditional expectation you mentioned. While I'm aware of measure theory, I haven't yet had a serious look at that definition yet. Which I'll do now.