Timeline for Continuity of the densities of a stochastic process
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 26, 2020 at 15:04 | vote | accept | fsp-b | ||
| Aug 26, 2020 at 15:00 | vote | accept | fsp-b | ||
| Aug 26, 2020 at 15:04 | |||||
| Aug 26, 2020 at 14:40 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 26, 2020 at 14:23 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Aug 26, 2020 at 14:20 | comment | added | Iosif Pinelis | @fsp-b : I don't think Kolmogorov-type conditions will help. If needed, you can make $X_t$ even closer to $X_0$ (for small $t$) by replacing $\sin\frac xt$ by something like $\sin(xe^{1/t})$ or $\sin(x\exp(e^{1/t}))$, with very high frequencies near $t=0$. | |
| Aug 26, 2020 at 14:00 | comment | added | fsp-b | Many thanks for your answer, Iosif Pinelis. I accept your counterexample (and thus also the one that mike gave in his answer). I would hope though that, maybe, continuity of $(t,x)\mapsto p_t(x)$ could be achieved by imposing sufficient growth conditions on the differences $\mathbb{E}[|X_t-X_s|^\alpha]$ (akin to Kolmogorov's continuity theorem). Do you have any ideas how sufficient `uniformity' conditions of this sort might look like? | |
| Aug 26, 2020 at 13:52 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |