View Single Post

 Quote by kai_sikorski 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 $\operatorname{E}(X_i|\mathcal{F}_i):\Omega \to \mathbb{R}$ is not a random variable but $\operatorname{E}(X_i|\mathcal{F}_{i-1}):\Omega \to \mathbb{R}$ is a random variable, although it has much less uncertainty than $X_i$.