Timeline for consequence of "the best coupling" of two SDEs with different diffusion matrices
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 16, 2021 at 17:56 | vote | accept | Fei Cao | ||
| Jul 16, 2021 at 10:10 | answer | added | Nawaf Bou-Rabee | timeline score: 1 | |
| Jul 15, 2021 at 22:32 | comment | added | Fei Cao | @NawafBou-Rabee Thank you! But there is a mistake: $X_t^2 = X_0^2 + \sigma_2 U(a_1,a_2) B_t^1$ instead of $X_t^2 = X_0^2 + U(a_1,a_2) B_t^1$. Also, it is hard to "simplify" $\mathbb{E}[ | X_t^1 - X_t^2|^2 ]$ and compare with the corresponding bound obtained via the synchronous coupling... | |
| Jul 15, 2021 at 19:25 | comment | added | Fei Cao | @NawafBou-Rabee thank you very much. But in my original post above, there are no constants such as $C_1$ and $C_2$, I am wondering where do these constants come from... | |
| Jul 15, 2021 at 16:56 | comment | added | Fei Cao | @NawafBou-Rabee thanks! But what are $C_1$ and $C_2$? It seems that it appears out of the blue. I also have no clue as to why you mentioned that "This result is straightforward to prove" | |
| Jul 15, 2021 at 15:01 | comment | added | Nawaf Bou-Rabee | As the authors state, $W_2(\mathcal{L}(X_t^1),\mathcal{L}(X_t^2))^2$ is precisely a "dynamical version" (meaning that it involves time) of the standard 2-Wasserstein distance between two multivariate normal distributions: en.wikipedia.org/wiki/Wasserstein_metric#Normal_distributions with $\mu_1 = X^1_0$, $\mu_2=X^2_0$, $C_1 = t a_1$ and $C_2 = t a_2$. This result is straightforward to prove. | |
| Jul 15, 2021 at 5:39 | history | asked | Fei Cao | CC BY-SA 4.0 |