Timeline for Tanaka stochastic differential equation and Kolmogorov equation
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 19, 2012 at 19:55 | comment | added | Jon | @GeorgeLowther: You got it! It is a Fokker-Planck equation I would like to see and I was not able to read from standard literature. Please, put it as an answer and I will accept it. Thanks a lot! | |
| Jul 19, 2012 at 18:16 | comment | added | George Lowther | I mean, at zero (not time 0). | |
| Jul 19, 2012 at 18:15 | comment | added | George Lowther | @Jon: I don't understand what you mean, what does "exist a kind of diffusion equation" mean? Do you mean a backwards equation/Fokker-Planck equation. There does, but it's a little different from the backward equations you get with smooth diffusion coefficient, as you have to enforce continuity of the derivative at time 0. | |
| Jul 19, 2012 at 10:52 | comment | added | Jon | @George Lowther: Thanks, but the question is, does it exist a kind of diffusion equation also in this case? | |
| Jul 19, 2012 at 10:37 | comment | added | George Lowther | Your SDE will have a unique weak solution which is a one dimensional diffusion with speed measure $(a{\rm sgn}(x)+b)^{-2}dx$ (as long as ${\rm sgn}(0)$ is defined appropriately). | |
| Jul 18, 2012 at 9:48 | history | edited | Jon | CC BY-SA 3.0 |
Fixed a mistake
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| Jul 18, 2012 at 9:48 | comment | added | Jon | @Didier Piau: Thanks, I will fix this. | |
| Jul 18, 2012 at 8:48 | comment | added | Did | @TheBridge is right, naturally: sign(X) is not sign(W). | |
| Jun 13, 2012 at 16:37 | comment | added | Jon | @The Bridge: There is such a big difference that I cannot see it and this is not the remark by Ilya at all. As you can note, if I take the two constants to be $a=1$ and $b=0$ you are back to Tanaka sde otherwise it is just the sum of a Tanaka plus a normal Wiener. | |
| Jun 13, 2012 at 15:55 | comment | added | The Bridge | @ Jon : a remark as noted by Ilya, your sde is not Tanaka's sde, the right equation is $dX_t=sign(X_t)dW_t$. Best regards. | |
| Jan 27, 2012 at 13:25 | comment | added | Jon | @Ilya: Thank you. Indeed, it was difficult to identify the very nature of this process. | |
| Jan 27, 2012 at 12:22 | comment | added | SBF | Well, it's not quite an SDE in the sense that the right-hand side does not depend on $X$, so you can write the explicit solution $$ X_t = X_0+\int\limits_0^t[a\sign W_s +b ]\mathrm dW_s. $$ On the other hand, the Kolmogorov equation is defined only for Markov processes, I guess. I am not sure that the process $X$ is Markov. | |
| Jan 19, 2012 at 9:42 | history | asked | Jon | CC BY-SA 3.0 |