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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?

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  • $\begingroup$ Have you taken a look at Jacod and Shiryaev's book "Limit Theorems for Stochastic Processes", they give some conditions on the characteristic functions for this to happen if I remember well. Regards $\endgroup$
    – The Bridge
    Commented Sep 24, 2018 at 8:31
  • $\begingroup$ Abit but I am not sure what part to look at... $\endgroup$
    – ABIM
    Commented Sep 24, 2018 at 14:12
  • $\begingroup$ A first observation in your context is that you can omit every part of characteristic functions associated to jumps, leaving two conditions on those functions to check, but you still have to determine the characteristic functions of your SDEs. $\endgroup$
    – The Bridge
    Commented Sep 24, 2018 at 14:23
  • $\begingroup$ Do you know the relevant part of the book to take a look at? $\endgroup$
    – ABIM
    Commented Sep 24, 2018 at 14:25

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Indeed as mentioned in the comments there is a stability of SDEs eg. 6.9 Theorem in "Limit Theorems for Stochastic Processes". enter image description here

See also here: "Stability of strong solutions of stochastic differential equations".

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