C.1 Proof of Theorem C.2
We will use the following lemma in the proof of Theorem C.2 .

Suppose that W\in D([0,1],\mathbb{R}^{r}) has finitely many jumps. Let X,\tilde{X}\in D([0,1],\mathbb{R}^{d}) be given by

\mathrm{d}X=b(X)^{-}\mathrm{d}W,\quad\mathrm{d}\tilde{X}=b(\tilde{X})\diamond\mathrm{d}W,\quad X(0)=\tilde{X}(0)=x.

Then

\|X-\tilde{X}\|_{p-var}\leq K\|b\|_{\mathrm{Lip}}\|b\|_{\infty}\sum_{t}|W(t)-W(t-)|^{2},

where K depends on \|b\|_{C^{\gamma}},\|W\|_{p\text{-var}},\gamma,p , and the sum is over all jump times t of W .

We are now ready to prove Theorem C.2 .

Let \tilde{X}_{n} be the solution to the Marcus RDE

\mathrm{d}\tilde{X}_{n}=b(\tilde{X}_{n})\diamond\mathrm{d}W_{n},\tilde{X}_{n}(0)=x_{n}.

The continuity of the solution map for generalised geometric RDEs gives that

\alpha_{p-var}(((W_{n},\tilde{X}_{n}),\phi_{b}),((W,X),\phi_{b}))=0

Then let \overline{X}_{n} be the solution to the Marcus RDE

\mathrm{d}\overline{X}_{n}=b_{n}(\overline{X}_{n})\diamond\mathrm{d}W_{n},\quad\overline{X}_{n}(0)=x_{n}.

On any subinterval [s_{1},s_{2}] , we can compare solutions to the Marcus RDE with b_{n} and b by using Proposition B.1 :

\|\overline{X}_{n}-\tilde{X}_{n}\|_{p-var} = \left\|\int_{0}^{.}(b_{n}(\overline{X}_{n})-b(\tilde{X}))\mathrm{d}W^{l_{r}}\right\|_{p-var} \leq \left\|\int_{0}^{.}(b_{n}(\overline{X}_{n})-b(\overline{X}_{n}))dW^{l_{r}}\right\|_{p-var}+\left\|\int_{0}^{.}(b(\overline{X}_{n})-b(\tilde{X}_{n}))dW^{l_{r}}\right\|_{p-var} \leq \|b_{n}-b\|_{\infty}\|W^{l_{r}}\|_{p-var}+C_{1}(\|b\|_{Lip}\|\overline{X}_{n}-\tilde{X}_{n}\|_{p-var}+|b(\overline{X}_{n})-b(\tilde{X}_{n})|(s_{1}))\|W^{l_{r}}\|_{p-var} = \|b_{n}-b\|_{\infty}\|W\|_{p-var}+C_{1}(\|b\|_{Lip}\|\overline{X}_{n}-\tilde{X}_{n}\|_{p-var}+|b(\overline{X}_{n})-b(\tilde{X}_{n})|(s_{1}))\|W\|_{p-var},

where the last inequality is due to Lemma C.1 .
If we choose a subdivision 0=s_{0}<s_{1}<\ldots<s_{n-1}<s_{n}=1 of the interval [0,t] such that for any i=0,1,\ldots,n-1 ,

C_{1}\|b\|_{Lip}\|W\|_{p-var,[s_{i},s_{i+1}]}\leq c<1,

then on the interval [0,s_{1}] ,

\|\overline{X}_{n}-\tilde{X}_{n}\|_{p-var,[0,s_{1}]} \leq\frac{\|b_{n}-b\|_{\infty}\|W\|_{p-var,[0,s_{1}]}+C_{1}\|W\|_{p-var,[0,s_{1}]}|b(\overline{X}_{n})-b(\tilde{X}_{n})|(0)}{1-C_{1}\|b\|_{Lip}\|W\|_{p-var,[0,s_{1}]}} \leq\|b_{n}-b\|_{\infty}\|W\|_{p-var,[0,s_{1}]}/(1-c).

Similarly on the interval [s_{1},s_{2}] ,

\|\overline{X}_{n}-\tilde{X}_{n}\|_{p-var,[s_{1},s_{2}]} \leq \frac{\|b_{n}-b\|_{\infty}\|W\|_{p-var,[s_{1},s_{2}]}+C_{1}\|W\|_{p-var,[s_{1},s_{2}]}|b(\overline{X}_{n})-b(\tilde{X}_{n})|(s_{1})}{1-C_{1}\|b\|_{Lip}\|W\|_{p-var,[s_{1},s_{2}]}} \leq \left(\|b_{n}-b\|_{\infty}\|W\|_{p-var,[s_{1},s_{2}]}+C_{1}\|W\|_{p-var,[s_{1},s_{2}]}\|b\|_{Lip}\|\overline{X}_{n}-\tilde{X}_{n}\|_{p-var,[0,s_{1}]}\right)/(1-c) \leq \|b_{n}-b\|_{\infty}\|W\|_{p-var,[s_{1},s_{2}]}/(1-c)+C_{1}\|b\|_{Lip}\|b_{n}-b\|_{\infty}\|W\|_{p-var,[0,s_{1}]}\|W\|_{p-var,[s_{1},s_{2}]}/(1-c)^{2}.

It is not hard to obtain by induction that there exists C(b,W)>0 such that

\|\overline{X}_{n}-\tilde{X}_{n}\|_{p-var,[0,1]}\leq C(b,W)\|b_{n}-b\|_{\infty}.

Therefore

\alpha_{p-var}(((W_{n},\tilde{X}_{n}),\phi_{b}),((W_{n},\overline{X}_{n}),\phi_{b_{n}}))=0

Since b_{n}\to b in C^{\gamma}(\mathbb{R}^{d},\mathbb{R}^{d\times r}) , \|b_{n}\|_{\mathrm{Lip}} and \|b_{n}\|_{\infty} are uniformly bounded independent of n . Then it follows from Lemma C.2 that

\|X_{n}-\overline{X}_{n}\|_{p-var}\leq K\|b_{n}\|_{\mathrm{Lip}}\|b_{n}\|_{\infty}\sum_{t}\|W_{n}(t)-W_{n}(t-)\|\to 0 as n\to\infty.

Then the same argument in [ 6 ] gives that for any p^{\prime}>p ,

\lim_{n\to\infty}\alpha_{p^{\prime}-var}(((W_{n},\overline{X}_{n}),l_{r+d}),((W_{n},X_{n}),l_{r+d}))=0

Recall also from the [ 6 ] that

\lim_{n\to\infty}\alpha_{p^{\prime}-var}(((W_{n},\overline{X}_{n}),\phi_{b_{n}}),(W_{n},\overline{X}_{n}),l_{r+d}))=0

Finally Theorem C.2 follows from combining ( 17 ), ( 18 ), ( 19 ) and ( 20 ). This completes the proof.
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