Given a complex variable $\alpha(t)$ following this stochastic differential equation $\dot{\alpha}(t)=-\kappa\alpha+\sqrt{\kappa n_{\rm th}} \eta(t)$, where $\eta(t)$ describes a Winer process, we can solve it numerically using Euler-Maruyama method
\begin{equation} \alpha(t+dt)=\alpha(t)-\kappa \alpha(t) dt + \sqrt{dt}\sqrt{\kappa n_{\rm th}} W(t) \end{equation} where $W(t)$ is sampled from a normal Gaussian distribution.
I am now a bit confused in solving the following coupled equations numerically: $\dot{x}(t)=-\gamma x(t) \dot{\alpha}(t)$. How should I use the Euler method here? Is it correct to use the following equation:
\begin{equation} \begin{split} &\alpha(t+dt)=\alpha(t)-\kappa \alpha(t) dt + \sqrt{dt}\sqrt{\kappa n_{\rm th}} W(t)\\ &x(t+dt)=x(t)-\gamma x(t) dt (-\kappa \alpha )-\gamma x(t)\sqrt{\kappa n_{\rm th}}\sqrt{dt}W(t) \end{split} \end{equation}
What worries me the most is the integration steps dt and the Wiener increments and that by plugging $\dot{\alpha}$ into $\dot{x}$, there must be some careful steps I am now aware.