Timeline for Blackbox Theorems
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 17, 2012 at 3:32 | comment | added | Nate Eldredge | Every book I've ever seen that proves the existence of BM goes on to prove it is nowhere differentiable. | |
| Jun 16, 2012 at 21:44 | comment | added | Alexander Shamov | The proof of continuity usually follows from the "Kolmogorov Criterion" ... which itself is a little technical, but nevertheless fairly intuitive chaining argument. | |
| Jun 15, 2012 at 23:54 | comment | added | Felipe Olmos | The proof of continuity usually follows from the "Kolmogorov Criterion": If there exists strictly positive constants $\varepsilon$, $p$ and $C$ such that $$\mathbb{E}|X_t - X_s|^p \leq C|t-s|^{1+\varepsilon}$$ then almost surely $X$ has a modification which has $\alpha$-Hölder continuous paths for any $\alpha \in (0,\frac{\varepsilon}{p})$ | |
| Jun 15, 2012 at 16:12 | comment | added | Zsbán Ambrus | Okay, you're right. I accept that you have to prove existence and continuity together. | |
| Jun 14, 2012 at 22:03 | comment | added | George Lowther | @Zsbán: Continuity of BM is part of the standard definition, so proving that is the same as proving that it exists. However, the proof that BM is almost-surely nowhere differentiable is probably less well known. | |
| Jun 14, 2012 at 9:49 | comment | added | Zsbán Ambrus | I think it's not so much the existence that's hard to prove but that it being continuous everywhere has probability 1. | |
| Jun 14, 2012 at 1:12 | history | answered | weakstar | CC BY-SA 3.0 |