Timeline for More natural example of measurable but not progressive process
Current License: CC BY-SA 4.0
13 events
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| Jan 6, 2023 at 14:57 | history | edited | tsnao | CC BY-SA 4.0 |
added p.s. with remark from dellacherie & meyer
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| Jan 3, 2023 at 0:20 | history | edited | tsnao | CC BY-SA 4.0 |
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| Jan 3, 2023 at 0:04 | comment | added | tsnao | Okay, the difference between two examples is that $\mathcal{G}_t = \mathcal{B} [ 0, t ]$ is much bigger than the natural filtration of $X_t ( \omega ) = 1_{t = \omega}$, which is just $\mathcal{H}_t$, hence the analogy of the second example is not valid. | |
| Jan 2, 2023 at 21:03 | comment | added | tsnao | @IosifPinelis, I'm clearly missing something... A singleton in the first example is a preimage of $X_t$, right? What is the difference then? | |
| Jan 2, 2023 at 21:00 | comment | added | Iosif Pinelis | No, not of the same nature. | |
| Jan 2, 2023 at 20:59 | comment | added | tsnao | @IosifPinelis, I mean, isn't the sigma-algebra generated by singletons in the first example of exactly the same nature? | |
| Jan 2, 2023 at 20:56 | comment | added | tsnao | @IosifPinelis, shouldn't it only work if $X$ has some continuity (right-continuity or at least separability for instance)? | |
| Jan 2, 2023 at 20:21 | comment | added | Iosif Pinelis | Concerning your modification, if $\mathcal{H}_t$ is generated by all finite subfamilies of $(X_s)_{s \in [0, t]}$, then $\mathcal{H}_t$ is the same as $\sigma$-algebra as the one generated by all $X_s$ with $s \in [0, t]$. Your first example seems fine to me. | |
| Jan 2, 2023 at 19:28 | history | edited | tsnao | CC BY-SA 4.0 |
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| Jan 2, 2023 at 19:15 | history | edited | tsnao | CC BY-SA 4.0 |
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| Jan 2, 2023 at 18:57 | history | edited | tsnao | CC BY-SA 4.0 |
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| Jan 2, 2023 at 18:51 | history | edited | tsnao | CC BY-SA 4.0 |
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| Jan 2, 2023 at 18:40 | history | answered | tsnao | CC BY-SA 4.0 |