How to match this SDE with Ito's formula

Question

I have a SDE of the following form

$\frac{dZ}{dt}=-\lambda Z+\sigma\alpha_t\beta$, where $\sigma$ being the standard deviation and $\alpha_t$ is the stochastic parameter. $\beta$ is deterministic. I have two main questions:

1. with $\alpha$ being stochastic, does it mean $Z$ also becomes stochastic because of the presence of $\alpha$?

2. I am trying to match this SDE with the standard one a.k.a $dX_t=\mu_tdt+\sigma_tdB_t$.

My attempt is as follows:

I can write my SDE as: $dZ=-\lambda Zdt+\sigma\alpha_t\beta dt$
We know that $dt=dB_t^2$ through Ito's result. so I can rewrite the above as $dZ=-\lambda Zdt+\sigma\alpha_t\beta dB_t^2$

Can I say that I have achieved the general form now with: $\mu_t=-\lambda Z$ and $\sigma_t=\sigma\alpha_t\beta dB_t$?

Or I need to utilize the Ito's lemma here? Basically my final goal is to achieve a result for $E[Z]$.

Answer 1

Too long for a comment:

$$Z_t=e^{-\lambda t}(Z_0+\sigma\beta\textstyle\int_0^te^{\lambda s}\alpha_s\,ds)\,,\quad E[Z_t]=e^{-\lambda t}(Z_0+\sigma\beta\textstyle\int_0^te^{\lambda s}E[\alpha_s]\,ds).$$