Timeline for Fokker-Planck equation for a truncated process
Current License: CC BY-SA 3.0
8 events
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| Nov 8, 2017 at 13:49 | comment | added | kenneth | I missed reading the claim of $p^0$ as the density (without proof) in the book and I agree with you that $p^0$ is $m$ in this post. Thanks. In that example, $b= 0$ and $G$ is equal to its adjoint $G^*$, therefore it partially confirms that the solution of the above Cauchy-Dirichlet PDE actually provides the density. But it is still desired for the precise statement for a general Markov diffusion. | |
| Nov 8, 2017 at 12:55 | comment | added | Kore-N | The same author addresses similar problems also in "Diffusions and Elliptic Operators". | |
| Nov 8, 2017 at 12:53 | comment | added | Kore-N | In my opinion it addresses exactly your issue. Indeed the $p^0$ he finds is the $m$ in your notation. Since he has an explicit description of $p^0$ in terms of a series expansion you can directly see that the boundary conditions are verified (and maybe you can generalize this series expansion to other situations). You also have the stochastic representation as $m(t,y) = \mathbb{P}(Y_t = y, t < T),$ which tells you that up to a normalizing constant $m(t, \cdot)$ is the transition probability conditioned on the hitting time being larger than $t$ | |
| Nov 8, 2017 at 12:00 | comment | added | kenneth | I've checked Bass's book, but it does not seem to directly answer this question. Especially, why the boundary shall be set zero is not explained. | |
| Nov 7, 2017 at 12:40 | comment | added | Kore-N | Yes, I agree with you, the second statement is supposed to be the answer to your question, in the case $b=0, \sigma =1$. The first statement was just to make things clear. If you want a reference, you might be interested in Section 40.3 of "Stochastic Processes" by Richard Bass. | |
| Nov 7, 2017 at 2:40 | comment | added | kenneth | Thanks for your reply. I agree with your first comment that the equation does not provide any probability of absorbing states, see the added remark. The second statement of your answer seems to me to support the equation itself. | |
| Nov 6, 2017 at 16:00 | history | edited | Kore-N | CC BY-SA 3.0 |
added 354 characters in body
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| Nov 6, 2017 at 15:54 | history | answered | Kore-N | CC BY-SA 3.0 |