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Let you have the following mean reverting process:

$\text{d}x_{t}=a(\theta-x_{t})\text{d}t$,

where the diffusion term is absent, that is this process is not stochastic.

Let you know the value of $\theta$.

You also know that at time $t=T$ it must be $x_{T}\simeq\theta$.

(That is when $|x_{T}-\theta|$ is so small to be negligible because $x_{t}=\theta$ when $t\rightarrow\infty$).

Does any closed form and/or a proxy of $a(\theta,T)$ exist?

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  • $\begingroup$ Could it be the $\beta$ of a log-regression $x_{t}\sim \log(t)$? $\endgroup$ – Lisa Ann May 16 '13 at 17:30
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    $\begingroup$ That's a linear ODE with a closed form solution. Have you tried using that? Solve it with separation of variables. $\endgroup$ – quasi May 16 '13 at 17:32
  • $\begingroup$ @quasi: perhaps, you can put this comment as an answer (I would upvote) $\endgroup$ – Ilya May 21 '13 at 11:30

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