Timeline for Identify two continuous martingales in law as time-changed Brownian motions
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| May 17, 2023 at 7:37 | comment | added | Fawen90 | @NateRiver We only have the identification in law between $X$ and $B_H$, where $B, H$ are defined in some abstract probability space (that may have nothing to do with $X$), so in generally we cannot claim $h=\alpha$ | |
| May 17, 2023 at 0:07 | comment | added | Nate River | I may be mistaken, but do we not have $H_t = \langle X, X \rangle_t = \int_0^t \alpha_s^2 ds$, so $h_s = \alpha_s$? And thus $G_t = \int_0^t \beta_s^2 ds = \int_0^t \text{max} (\alpha_s, 1) ds = \int_0^t g_s^2 ds$? | |
| Apr 13, 2023 at 6:58 | history | edited | Fawen90 | CC BY-SA 4.0 |
deleted 56 characters in body; edited title
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| Apr 11, 2023 at 15:45 | history | edited | YCor |
edited tags
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| Apr 11, 2023 at 15:24 | history | edited | Michael Hardy | CC BY-SA 4.0 |
somewhat better MathJax usage
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| Apr 11, 2023 at 12:59 | comment | added | Fawen90 | Let us continue this discussion in chat. | |
| Apr 11, 2023 at 12:58 | comment | added | Thomas Kojar | But the equality in law is about integrals not about realizations. Whereas you are asking to have the same Brownian motion. | |
| Apr 11, 2023 at 12:57 | comment | added | Thomas Kojar | not it is not even true then. | |
| Apr 11, 2023 at 12:57 | comment | added | Fawen90 | @ThomasKojar What I look for is just a equality in law | |
| Apr 11, 2023 at 12:56 | comment | added | Fawen90 | @ThomasKojar I don's really see what you mean. For the deterministic case, my desired equality is clearly true | |
| Apr 11, 2023 at 12:20 | comment | added | Thomas Kojar | But that doesn't make sense really. The distribution statement is about probability measures i.e. comparing integrals over all omega. Whereas you are asking for having the same realization omega i.e. you are asking for almost sure statement. If it was true, then Dubin's Schwarz would be true almost surely. | |
| Apr 11, 2023 at 12:18 | comment | added | Fawen90 | @ThomasKojar For two different Brownian motions the statement is clearly true. Here the desired identification is just in law, so I wish to known whether we may take the same Brownian motion | |
| Apr 11, 2023 at 12:07 | comment | added | Thomas Kojar | To be clear the two Brownian motions will be different because Dubins-Schwarz is only a statement on distribution not almost surely i.e. $Law(X)=B_{H}$ and $Law(Y)=\tilde{B}_{G}$. | |
| Apr 11, 2023 at 7:10 | history | asked | Fawen90 | CC BY-SA 4.0 |