Timeline for Fokker-Planck equation for a truncated process
Current License: CC BY-SA 3.0
14 events
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| Dec 13, 2017 at 15:35 | history | edited | Nawaf Bou-Rabee | CC BY-SA 3.0 |
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| Nov 9, 2017 at 2:50 | comment | added | kenneth | Yes, That's true | |
| Nov 9, 2017 at 1:42 | comment | added | Nawaf Bou-Rabee | They are both arithmetic brownian motions: just flip the sign of b in Z to obtain the corresponding quantity for X. | |
| Nov 9, 2017 at 1:20 | comment | added | kenneth | To me, $p$ is the density of $Z$, not $X$. Am I missing something? | |
| Nov 8, 2017 at 23:01 | comment | added | Nawaf Bou-Rabee | Doesn’t the addendum show that? | |
| Nov 8, 2017 at 22:59 | comment | added | kenneth | Thanks for your added comment. Indeed, shall we directly show that why $\mathbb E_y m_0(Z_t)$ is the density of $X_t$ at $dy$, with $m_0$ fixed as the law of $X_0$? | |
| Nov 8, 2017 at 15:20 | history | edited | Nawaf Bou-Rabee | CC BY-SA 3.0 |
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| Nov 8, 2017 at 15:08 | comment | added | Nawaf Bou-Rabee | @kenneth I added some background material. I hope it helps. | |
| Nov 8, 2017 at 15:08 | history | edited | Nawaf Bou-Rabee | CC BY-SA 3.0 |
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| Nov 8, 2017 at 13:36 | comment | added | kenneth | Indeed your suggestion is quite fresh to me. But I do not see how this argument implies that the PDE solution $m$ is not the desired density. For instance, if we consider the solution $m$ of the first Fokker-Planck equation for the density of $X$ in the above, one can write $m(t, y) = \mathbb E_y [ m_0(Z_t) ]$ by Feymann-Kac formula. However, I do not see how $m$ relates to the density of $X_t$ from this representation, although it's the fact. | |
| Nov 8, 2017 at 13:02 | comment | added | Nawaf Bou-Rabee | @kenneth A local solution of the PDE in m does have a stochastic representation, which is given in my answer. It, however, has no immediate relation to the representation you expect. | |
| Nov 8, 2017 at 11:58 | comment | added | kenneth | Thanks for your idea. I could not be completely convinced by this argument. At least I believe the equation itself is correct in $(0, \infty) \times (0, 1)$ for its local property of the diffusion. | |
| Nov 7, 2017 at 2:38 | comment | added | kenneth | Thanks for your reply, let me think about it and reply you later. | |
| Nov 6, 2017 at 16:34 | history | answered | Nawaf Bou-Rabee | CC BY-SA 3.0 |