Timeline for Characterization of martingale diffusions ending in $\{-1,1\}$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Nov 18, 2021 at 21:03 | vote | accept | GJC20 | ||
| Nov 8, 2021 at 18:55 | history | edited | GJC20 | CC BY-SA 4.0 |
added 208 characters in body
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| Nov 7, 2021 at 22:30 | comment | added | Mateusz Kwaśnicki | I would say this is more related to existence (of a non-trivial solution) rather than uniqueness. More specifically, this is related to Feller's classification of 1-D diffusion processes. This is too long for a comment, I elaborated a bit in an answer below. | |
| Nov 7, 2021 at 22:28 | answer | added | Mateusz Kwaśnicki | timeline score: 2 | |
| Nov 7, 2021 at 21:14 | comment | added | GJC20 | @MateuszKwaśnicki Thanks a lot for the quick answer. I suppose for the homogenous case, your condition is related to the uniqueness of the SDE $dX_t=a(X_t)dW_t$ with $X_0=x\in [-1,1]$. Could you please specify the reasoning? Thank you very kindly | |
| Nov 7, 2021 at 20:58 | comment | added | Mateusz Kwaśnicki | For the time-homogeneous case (when $a(t,x)$ does not really depend on $t$), the answer is fairly simple: $a(t,x)=a(x)$ is fine if (a) $a(x) = 0$ when $|x| \geqslant 1$, and (b) $a(x) > 0$ on every compact interval in $(-1, 1)$. (Condition (b) can be relaxed to $a > 0$ in $(-1, 1)$ with $1/a^2$ locally integrable, I suppose.) However, I have no idea what the answer could possibly be in the general case. | |
| Nov 7, 2021 at 20:15 | history | asked | GJC20 | CC BY-SA 4.0 |