In short form: Does a general solution to d$\dot{m}(t,\theta) = [\alpha_t \dot{m}(t, \theta)+\beta_tm(t,\theta)]dt+\sigma_t dW_t$ exist? The dot marks differentiating with respect to $\theta$. I know that a linear SDE with additive noise of the form $dX_t=\beta_tX_tdt+\sigma_tdB_t$, where $\beta, \sigma$ are deterministic functions, has an explicit solution. But, in this example, I am not sure how to compute one as the SDE includes both $m$ and $\dot{m}$.
In long form: I am reading Chapter 6 of the book "Identification of Dynamical Systems with Small Noise" by Kutoyants which deals with Partially Observed Systems. We have the system of equations \begin{align} dX_t=f_t(\theta)Y_tdt+\varepsilon dW_t \\ dY_t=b_t(\theta)Y_tdt+\varepsilon \sigma_t(\theta)dV_t\end{align} for $0\leq t \leq T$, $W_t, V_t$ are independent Wiener processes, $f, b, \sigma$ continuous and bounded with their first two derivatives wrt $\theta$. The component $X_t$ is observed, but $Y_t$ is not. We want to construct estimators for the unknown parameter $\theta$. By the innovation theorem, the process $X_t$ also admits the representation as a diffusion-type process $dX_t=S_t(\theta,X)dt+\varepsilon d\bar{W}_t$, where $S_t(\theta,X)=f_t(\theta)m_t(\theta)$ and $m_t(\theta)$ is the conditional expectation of $Y_t$ wrt $X_t$, i.e. $m_t(\theta)=\mathbb{E}[Y_t|X_s, 0 \leq s \leq t]$.
Kutoyants begins by showing that the family of measures $\{ P_\theta^{(\varepsilon)}, \theta \in \Theta \}$ is locally asymptotically normal (LAN) with normalizing matrix $\varphi_\varepsilon (\theta)=\varepsilon I(\theta)^{-1/2}$ and vector $\Delta (\theta,x)=I(\theta)^{-1/2} \int_0^T [\dot{f_t}(\theta)y_t(\theta)+f_t(\theta)\dot{y_t}(\theta)]d\bar{W}_t$. The steps up to "from the linearity" are clear, but then it says:
and I have no idea how to continue here. The same problem, namely how I solve the above-mentioned SDE, also appears in the further development again and again, so I would be very grateful for some hints or tips!