I've been looking for a while an cannot seem to find the answer to this question:
If $Y_n$ is a sequenence of semi-martingales solving the sequence of stochastic differential equations: $$ dY_n(t) = \mu_n(t,Y_n(t))dt + \Sigma_n(t,Y_n(t))dW(t) $$ where $\mu$ and $\Sigma$ are simple adapted processes to the filtration of the $d$-dimensional Brownian motion $W_t$ then is $\mu_n,\Sigma_n$ converge to adapted processes $\mu$ and $\Sigma$ does $\lim_{n \mapsto \infty} Y_n(t)$ solve the Stochastic Differential Equation: $$ dY(t) = \mu(t,Y(t))dt + \Sigma(t,Y(t))dW(t)? $$