Timeline for PDE for the probability of Brownian motion staying in an area (reference request)
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Mar 31 at 23:10 | comment | added | tsnao | I still cannot reproduce your argument for why $u(t-s, B_{s \wedge \tau} + x)$ is a martingale. I'm probably doing something stupid, but what you wrote doesn't seem enough... | |
| Mar 31 at 11:55 | comment | added | Kostya_I | @tsnao, I added some details, hope that helps! In the case $A$ is not compact, the solution to the parabolic problem is not unique, so you need to pick a bounded one (then the above argument shows that a bounded solution is unique). | |
| Mar 31 at 11:50 | history | edited | Kostya_I | CC BY-SA 4.0 |
details added
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| Mar 31 at 11:14 | comment | added | tsnao | I don't quite see how this proof works. First, showing that if $\hat{u}$ is a solution of heat equation $\implies \hat{u} ( t - s, B_{t \wedge \tau} + x )$ is a martingale is straightforward, I agree. But how do we show that $u(t-s, B_{s \wedge \tau} + x )$ with $u$ from my post is a martingale? Second, how do boundary conditions come into play here? And third, it seems that your proof requires $\tau$ to be bounded, but does it also work if $A$ is not compact? | |
| Mar 31 at 6:43 | history | answered | Kostya_I | CC BY-SA 4.0 |