Suppose we have the following two stochastic differential equations for $x_0$ and $x$ respectively \begin{align} dx_0 &= -k_0(t)(x_0-1)dt+\eta_0(t) x_0\,dB \tag1\\ dx &= -(k_0(t)+\epsilon k_1(t))(x-1)dt+(\eta_0(t)+\epsilon \eta_1(t)) x\,dB \tag2 \end{align} with initial condition $x(t=0)=x_0(t=0)$, where $\epsilon>0$ is a constant parameter and $k_i(t)$ and $\eta_i(t)$ for $i\in\{0,1\}$ are $t$-dependent functions. Each SDE thus has a unique solution for any given initial value.

Let $$x=x_0+\epsilon y$$ and substitute it into Eq. (2) and collect up-to 1'st power the same power terms of $\epsilon$. $$(dx_0+k_0(t)(x_0-1)-\eta_0(t) x_0 dB)+\epsilon(dy+(k_0(t)y+k_1(t)(x_0-1))\,dt-(\eta_0(t)y+\eta_1(t)x_0)dB)+O(\epsilon^2)=0. \tag3$$ The term in the first parenthesis vanishes due to Eq. (1). We set $$dx_1=-(k_0(t)x_1+k_1(t)(x_0-1))dt+(\eta_0(t)x_1+\eta_1(t)x_0)dB. \tag4$$ with initial condition $x_1(t=0)=0$.

$(\text{Eq}.(3)-\text{Eq}.(1))/\epsilon$ gives $$dy=-(k_0y+k_1(x_0-1)+\epsilon k_1y)\,dt+(\eta_0y+\eta_1x_0+\epsilon \eta_1y)\,dB. \tag5$$ $\text{Eq}.(5)-\text{Eq}.(4)$ gives $$dz = -(k_0z+\epsilon k_1y)\,dt+(\eta_0z+\epsilon \eta_1y)\,dB \tag6$$ where $z=y-x_1$.

**Question**: Does $z\rightarrow0$ as path/trajectory or function of time, in some sense, e.g. in distribution or probability, as $\epsilon\rightarrow0$?

We can use the Duhamel's principle to obtain an explicit Ito integral solution for Eq. (2) and convergence is clear and it is pathwise. But I would like to use this problem as a model for techniques that can be generalized to the case where the factors in front of $dt$ and $dB$ are Lipschitz continuous.