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I am currently engaging a research that would really use your help. I am considering add a brownian-type shock to a "fraction" $\theta \in [0,1]$, for example $$d\theta_{t} = \sigma \theta_{t} dB_{t}$$ where $\sigma$ is a constant.
However, the standard brownian motion does not necessary limit $\theta \in [0,1]$, I am just wondering if there is any stochastic process or transformation of Brownian motion truncation method that can make $\theta \in [0,1]$.

I think the simplest method is truncating directly, i.e. whenever the resulting $\theta>1$ (or smaller than $0$), we can make them truncated at $1$ or $0$, but I am pretty sure this process does not satisfy the standard properties of brownian motion (e.g. use Ito's lemma involving $\theta$), or does it?

Any help would be extremely appreciated!!

Thank you

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You can very well choose a form $$ d\theta_t = f(\theta_t) dB_t $$ where $f(0) = f(1) = 0$. This way, when you approach 0 or 1, the speed decreases you you can never get out of $[0,1]$.

For a documented example, look for the Wright-Fisher diffusion.

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  • $\begingroup$ Hi @mookid, Thank you for your reply, I am wondering if the following example works $$d\theta_{t} = \sigma \theta_{t} (1-\theta_{t})dB_{t}$$ Not 100% sure $\endgroup$ – EconGuy Oct 26 '14 at 22:51
  • $\begingroup$ It does!${}{}{}$ you can also have a drift part. If it is zero as well on the drift part, it remains in $[0,1]$. It may or may not be possible to arrive at the barrier in finite time. But if you attain the barrier you can't move any more. $\endgroup$ – mookid Oct 26 '14 at 22:51
  • $\begingroup$ It might be a stupid question but since theoretically $dB$ is not necessarily between 0 and 1, and if $dB = 10$, and $\theta = 0.5$, let $\sigma$ = 1, then we have $d\theta = 2.5$, I am just trying to think about discrete time analogue.. sorry it might sound stupid but I try to understand it, thank you! $\endgroup$ – EconGuy Oct 26 '14 at 23:00
  • $\begingroup$ remember that the trajectories of the Brownian motion are continuous! $\endgroup$ – mookid Oct 26 '14 at 23:03
  • $\begingroup$ You are welcome! $\endgroup$ – mookid Oct 26 '14 at 23:05

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