Timeline for Local martingale for a (two-dimensional) diffusion
Current License: CC BY-SA 4.0
6 events
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| May 26, 2023 at 3:23 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| May 23, 2023 at 18:27 | comment | added | Serguei Popov | @ThomasKojar $X$ is the Doob's $h$-transform (not with that $h$, though) of the Brownian motion (it's a 2-dimensional BM conditioned on not touching a bounded domain), and $h$ is some complicated thingy which involves the expected value of some function on the boundary of that domain wrt the entrance measure from $x$ there (by the BM). So I was a bit in doubt how to differentiate it correctly... On the other hand, that equality ${\bf E}_x (...) = h(x)$ is easy to obtain. But I've already figured out that one can insert $t\wedge \tau_r$ there instead of just $\tau_r$, thus solving my problem. | |
| May 23, 2023 at 16:33 | comment | added | Serguei Popov | @ThomasKojar it's fine to assume some regularity for $h$ (maybe it even has to be "nice" if $X$ is a "good" diffusion --- in the example I have in mind $f$ is an analytic function). Btw, I think I've already figured out how to circumvent my specific issue; but nevertheless I'm curious how can one pass from a "sequence of stopping times"-statement to a "fixed $t$"-statement. | |
| May 23, 2023 at 16:15 | comment | added | Thomas Kojar | But I think it is still interesting to see if we can apply Ito in the first place. That means we have to extract some regularity from the mean value property stated for $h$ in the spirit of using MVP results for elliptic equations like "a mean value property of elliptic equations..." ams.org/journals/proc/1967-018-06/S0002-9939-1967-0218747-X/… which will depend on the regularity of $f$. | |
| May 23, 2023 at 8:58 | history | edited | Serguei Popov |
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| May 23, 2023 at 8:45 | history | asked | Serguei Popov | CC BY-SA 4.0 |