Timeline for Ornstein-Uhlenbeck type process with thresholding
Current License: CC BY-SA 4.0
11 events
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| May 24, 2018 at 11:05 | comment | added | Iosif Pinelis | I think you should recheck your simulation at least for $\gamma\le1/2$. If $X$ takes a very large positive (say) value, then there will be a strong negative trend. | |
| May 24, 2018 at 3:11 | comment | added | Nick | Thank you! I ran some simulations. For $\gamma>1/2$, the behavior of $X$ does look like $\sigma B$. For $\gamma<1/2$, $X$ explodes. However, rather than oscillating between positive and negative values, $X$ explodes toward one of the directions. Some simulation paths explode toward $+\infty$, some toward $-\infty$, but they do not oscillate. | |
| May 23, 2018 at 20:44 | comment | added | Iosif Pinelis | I think, again in view of the law of the iterated logarithm, it will depend on whether $\gamma\le1/2$. If so, I think $X(t)$ will be oscillating between $(k+o(1))t^\gamma$ and $(−k+o(1))t^\gamma$ as $t\to\infty$. If $\gamma>1/2$, then I think $X$ will be similar to $\sigma B$. | |
| May 23, 2018 at 19:51 | history | edited | Nick | CC BY-SA 4.0 |
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| May 23, 2018 at 19:39 | history | edited | Nick | CC BY-SA 4.0 |
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| May 21, 2018 at 1:49 | comment | added | Nick | Thank you. Suppose I consider the following process instead:$$ dX(t) = - c\ \mbox{sgn}(X(t))\big[|X(t)|-c_1 t^{\gamma}\big]_+ dt + \sigma dB(t), \quad X(0)=0, $$ where $c_1>0$, I wonder if anything can be said about the behavior of $X(t)$ with respect to different value of $0 < \gamma\leq 1$? | |
| May 20, 2018 at 21:32 | comment | added | Iosif Pinelis | I think $X(t)$ will be oscillating between $(k+o(1))\sqrt t$ and $(-k+o(1))\sqrt t$ as $t\to\infty$, for some real $k>0$. So, it seems that no stationary distribution exists. Also, in view of the law of the iterated logarithm for the Brownian motion, with probability $1$ the paths of $X$ will be substantially different from those of the Brownian motion. Also, there seems to be no reason to believe that there exists a closed-form solution to this SDE. | |
| May 20, 2018 at 21:17 | history | edited | Nick | CC BY-SA 4.0 |
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| May 20, 2018 at 21:07 | history | edited | Nick | CC BY-SA 4.0 |
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| May 20, 2018 at 21:05 | review | First posts | |||
| May 20, 2018 at 21:16 | |||||
| May 20, 2018 at 21:01 | history | asked | Nick | CC BY-SA 4.0 |