I have a stochastic differential equation for a position $x$ with respect to time $t$ $$ \frac{\mathrm{d} x}{\mathrm{d} t} = F[x(t), t] + \beta^\star(t) $$ where $F$ is a linear function of $x$ and $t$ (more precicely $F = a t - b x$) and interestingly $\beta^\star$ is a Brownian noise term with variance proportional to the velocity $v(t) = \frac{\mathrm{d} x}{\mathrm{d} t}$.
I would like to rewrite this in terms of the velocity by differentiating with respect to time $t$ $$ \frac{\mathrm{d} v}{\mathrm{d} t} = \frac{\mathrm{d} F}{\mathrm{d} t} + \frac{\mathrm{d} \beta^\star(t)}{\mathrm{d} t} $$ where naturally $\frac{\mathrm{d} F}{\mathrm{d} t} = a - b v$, but I'm don't know how to deal with the variance of the last term. It should be a white noise term, but with which variance?
I tried to apply the Itô differential here but didn't get far as the idea seems to be that the variance of $\beta^\star$ should be constant. What should I do here?
EDIT: So essentially my first equation (I changed to the notation $\beta^\star$ for the Brownian noise with variance $c v$) could be rewritten to $$ v = F[x(t), t] + \sqrt{c v} \beta = a t - b x + \sqrt{c v} \beta $$ where $c$ is a constant and $\beta$ Brownian noise term with unit variance. I would like to transform this into the Itô form $$ \mathrm{d} v = (a - b v) \mathrm{d} t + L[v(t), t] \mathrm{d} \beta $$ but don't know how to get to the function $L$.
User @gt6989b suggested that I consider $$ \mathrm{d} v = (a-bv) \mathrm{d} t + \frac{v}{Z} \mathrm{d} B_t $$ where the variance of $\mathrm{d} B_t$ is $Z^2$. I get the part that $\mathrm{d} B_t / Z$ would have variance 1 but don't understand how he has arrived to the $v$ in the last term. Comparing this to my equation would yield $$ L[v(t), t] \mathrm{d} \beta = \frac{v}{Z} \mathrm{d} B_t$$ which I think would imply that $L = v$?
HINT If the variance of $dB_t$ is $Z^2(t)$, then I would consider the SDE as $$ dv = (a - bv)dt + \frac{v}{Z(t)} dB_t $$ where the deterministic term would give you your equation without the white noise, the variance of $\frac{dB_t}{Z(t)}$ is $1$