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For each $n \in \mathbb{N}$ let $\mu_n(t,x)$ and $\sigma_n(t,x)$ be a sequence of processes adapted to the filtration of a d-dimensional Brownian motion $W(t)$: $$ dY_n(t) = \mu_n(t,Y_n(t))dt + \sigma_n(t,Y_n(t))dW(t) $$ such that $\mu_n \mapsto \mu$ in $L^2$ and $\sigma_n \mapsto \sigma$ in $L^2$.

Then (or under what conditions) does the solutions $Y_n$ to the above sequence of SDEs converge to the solution to the SDE

$$ dY(t) = \mu(t,Y(t))dt + \sigma(t,Y(t))dW(t)? $$

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You need to assume some structure conditions on $\mu$ and $\sigma$ these notes cover it on page 52 onwards.

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