Timeline for Convergence of the quadratic variation process
Current License: CC BY-SA 4.0
12 events
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| May 24, 2023 at 12:58 | comment | added | Nate River | In fact I just realised you can even take $X^n$, $X$ to be deterministic… just take any sequence $f_n$ of continuous functions with quadratic variations going to $\infty$ converging uniformly to $0$, and set $X_n = f_n, X = 0$ a.s. | |
| May 24, 2023 at 11:34 | comment | added | El_mago | The original problem I am working on, I have already explained here There I was suggested to use other methods to prove the questioned claim which I am grateful for. However, I still try to finish the idea I had. | |
| May 24, 2023 at 11:11 | comment | added | Nate River | Actually, i think this is still not possible even with the assumption that it be continuous. Vaguely, the idea is to replace the jumps in this example with small continuous segments with nontrivial quadratic variation. This will yield essentially the same counterexample. | |
| May 24, 2023 at 11:09 | comment | added | Nate River | Hmm, I don’t see how immediately. Perhaps you could create a separate thread to ask the modified question, maybe others can help. | |
| May 24, 2023 at 11:03 | comment | added | El_mago | Many thanks for your excellent counterexample. My questions were raised while I am trying to show that if $\lim_{n\to\infty}\mathbb{E}(|X^n_t-X_t|^2)$ and $\lim_{n\to\infty}\langle X^n\rangle=\infty$ then I hope to infer $\langle X\rangle=\infty.$ This seems to be still possible to prove. I am also additionally assuming that the $X^n$s have continuous sample paths. Do you have an idea how to prove this? | |
| May 24, 2023 at 10:16 | history | edited | Nate River | CC BY-SA 4.0 |
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| May 24, 2023 at 10:10 | history | edited | Nate River | CC BY-SA 4.0 |
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| May 24, 2023 at 10:02 | history | edited | Nate River | CC BY-SA 4.0 |
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| May 24, 2023 at 9:57 | history | edited | Nate River | CC BY-SA 4.0 |
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| May 24, 2023 at 9:56 | history | undeleted | Nate River | ||
| May 24, 2023 at 9:47 | history | deleted | Nate River | via Vote | |
| May 24, 2023 at 9:44 | history | answered | Nate River | CC BY-SA 4.0 |