Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a Lipschitz function from $\mathbb{R}^d$ to $\mathbb{R}$, also. Let $X^n_t$ solve the SDEs : $$ dX_t^n= b(X_t^n)dt+ a_n(X_t^n)dW_t, $$ where $W_t$ is a Brownian motion. Suppose furthermore that $X_t$ solves the SDE $$ dX_t= b(X_t)dt+ a(X_t)dW_t. $$

Under what (reasonable) conditions on $a_n$ is $\{X_t^n\}_{n \in \mathbb{N}}$ guaranteed to converge to $X_t$ in the *semimartingale topology?*