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
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.