Consider the two following real Stochastic Differential Equations (SDE) starting from the same initial condition: $$dx_t = f(x_t)dt + \sigma dB_t$$ $$dy_t = f(y_t)g_{\epsilon}(y_t)dt + \sigma dB_t$$

where $f$ and $g_{\epsilon}$ are such that there exists strong solutions to both SDEs (typically local Lipschitz assumptions on the coefficients).

We assume that $|1-g_{\epsilon}(y)|\leq \epsilon$ for all $y\in \mathbb{R}$.

I want to prove the following convergence: for all finite time $T>0$,

$$\lim_{\epsilon \to 0} \mathbb{E}\left[\sup_{0\leq t \leq T} |x_t-y_t|\right]=0$$

I would like to know how to prove it without a global Lipschitz assumption (think for instance that there may be some quadratic terms in $f$).

Can anyone explain me how to do it rigorously or point me to some article/book where it is already done ?

Thanks !