Timeline for Lower bound of $\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]$ (without observing history)
Current License: CC BY-SA 4.0
11 events
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| Jul 10, 2023 at 12:40 | history | edited | Fawen90 | CC BY-SA 4.0 |
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| Jul 10, 2023 at 11:13 | history | edited | Fawen90 | CC BY-SA 4.0 |
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| Jul 8, 2023 at 14:49 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Jul 7, 2023 at 19:56 | comment | added | Thomas Kojar | I didn't work through finding the particular pde-kernel/symbol for your case. It might be possible though as with Brownian bridge to come back to unconditional i.e. $X_{cond,t}=X_t-\frac{t}{T}X_{T}+x$. | |
| S Jul 7, 2023 at 19:52 | history | suggested | Pedro | CC BY-SA 4.0 |
Improve readability using `\mid` instead of `|`.
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| Jul 7, 2023 at 19:19 | comment | added | Fawen90 | @ThomasKojar Thanks for further explanation. Do you mind specifying $\mathbb P^x$? It seems to be the conditional expectation knowing $X_0=x$ for me, while my case is different. My probability is conditioning on the present, but the past | |
| Jul 7, 2023 at 15:55 | comment | added | Thomas Kojar | There they have a nice bound $$\mathbb{P}^x \left( \sup_{s \leq t} |X_s-x|\leq r \right) \geq 1-ct \sup_{|y-x| \leq r} \sup_{|\xi| \leq r^{-1}} |q(y,\xi)| \quad \text{for all $t \geq 0$}. $$ This is bad for large t. But I think the spirit of the approach of using the kernel to study the small-deviation might be useful for your problem. | |
| Jul 7, 2023 at 15:54 | comment | added | Thomas Kojar | sorry, actually I was thinking of something else. Another suggestion is to take the density route (assuming it has one) as done here math.stackexchange.com/questions/2948629/… | |
| Jul 7, 2023 at 15:26 | comment | added | Fawen90 | @ThomasKojar Thanks a lot for the reference. What is not clear to me is how to relate my process $X$ to a gaussian process. Could you please explain why $X$ is in this framework? | |
| Jul 7, 2023 at 14:08 | review | Suggested edits | |||
| S Jul 7, 2023 at 19:52 | |||||
| Jul 7, 2023 at 10:12 | history | asked | Fawen90 | CC BY-SA 4.0 |