\cr
    &=:&\mathcal{N}_1+\mathcal{N}_2+\mathcal{N}_3.\end{aligned}$$

The terms $\mathcal{N}_1$ and $\mathcal{N}_2$ can be estimated in the same way as the terms $\mathcal{M}_2$ and $\mathcal{M}_3$ in the proof of Lemma \[lemboundN\], using Lipschitz condition instead of the growth condition. This leads to the bounds $$\begin{aligned}
    \mathcal{N}_1+\mathcal{N}_2 \leq  C\int_{0}^{T}\mathbb{E}\big[\|u^{N,n}-u^{N,m}\|_{\alpha,t}^2\mathbf{1}_{D_{t}^{N,R}}\big]\mathrm{d}t.\end{aligned}$$

For $\mathcal{N}_3$, we have $$\begin{aligned}
    \mathcal{N}_3
    &\leq&C \mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t}\Big|\int_{0}^{t} S_{t-r} g(u^{N,n}_r) \mathrm{d}B^{H,N,n}_r-\int_{0}^{t} S_{t-r} g(u^{N,m}_r) \mathrm{d}B^{H,N,m}_r\cr
    &&-\int_{0}^{s} S_{s-r} g(u^{N,n}_r) \mathrm{d}B^{H,N,n}_r+\int_{0}^{s} S_{s-r} g(u^{N,m}_r) \mathrm{d}B^{H,N,m}_r\Big|(t-s)^{-1-\alpha}\mathrm{d}s\Big)^2\mathbf{1}_{D_{t}^{N,R}}\mathrm{d}t\Big]\cr
    &&+
    C\mathbb{E}\Big[\sup _{t \in[0, T]} \Big(\Big|\int_{0}^{t} S_{t-r} g(u^{N,n}_r) \mathrm{d}B^{H,N,n}_r-\int_{0}^{t} S_{t-r} g(u^{N,m}_r) \mathrm{d}B^{H,N,m}_r\Big|^2\mathbf{1}_{D_{t}^{N,R}}\Big)\Big]\cr
    &\leq&C \mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t}\frac{\big|\int_{s}^{t} S_{t-r} g(u^{N,n}_r) \mathrm{d}(B^{H,N,n}_r-B^{H,N,m}_r)\big|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathbf{1}_{D_{t}^{N,R}}\mathrm{d}t\Big]\cr
    &&+C \mathbb{E}\Big[\int_{0}^{T}\Big( \int_{0}^{t}\frac{\big|\int_{0}^{s}(S_{t-r}-S_{s-r}) g(u^{N,n}_r) \mathrm{d}(B^{H,N,n}_r-B^{H,N,m}_r)\big|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathbf{1}_{D_{t}^{N,R}}\mathrm{d}t\Big]\cr
    &&+C \mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t}\frac{\big|\int_{s}^{t} S_{t-r} (g(u^{N,n}_r)-g(u^{N,m}_r)) \mathrm{d}B^{H,N,m}_r\big|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathbf{1}_{D_{t}^{N,R}}\mathrm{d}t\Big]\cr
    &&+C \mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t}\frac{\big|\int_{0}^{s} (S_{t-r}-S_{s-r}) (g(u^{N,n}_r)-g(u^{N,m}_r)) \mathrm{d}B^{H,N,m}_r\big|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathbf{1}_{D_{t}^{N,R}}\mathrm{d}t\Big]\cr
    &&+C\mathbb{E}\Big[\sup _{t \in[0, T]} \Big|\int_{0}^{t} S_{t-r} g(u^{N,n}_r) \mathrm{d}(B^{H,N,n}_r-B^{H,N,m}_r)\Big|^2\mathbf{1}_{D_{T}^{N,R}}\Big]\cr
    &&+C\mathbb{E}\Big[\sup _{t \in[0, T]} \Big|\int_{0}^{t} S_{t-r} (g(u^{N,n}_r)-g(u^{N,m}_r)) \mathrm{d}B^{H,N,m}_r\Big|^2\mathbf{1}_{D_{T}^{N,R}}\Big]\cr
    &=:& \sum_{i=1}^{6}\mathcal{N}_{3i}.\end{aligned}$$

In the same way as for the term $\mathcal{M}_4$ in the proof of Lemma \[lemboundN\], we have $$\begin{aligned}
    \mathcal{N}_{31}+\mathcal{N}_{32} &\leq&
    C\mathbb{E}\Big[\int_{0}^{T}\big(\Lambda^{0,T}_{\alpha,B^{H,N,n,m}}\big)^2\Big(\int_{0}^t\frac{ r^{-\alpha}+(t-r)^{-\alpha}}{(t-r)^{\alpha}}(1+|u^{N,n}_r|)\mathrm{d}r\mathbf{1}_{D_{t}^{N,R}}\Big)^2\mathrm{d}t\Big]\cr
    &&+C\mathbb{E}\Big[\int_{0}^{T}\big(\Lambda^{0,T}_{\alpha,B^{H,N,n,m}}\big)^2\Big(\int_{0}^t \frac{\int_{0}^{r}\frac{|u^{N,n}_r-u^{N,n}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q}{(t-r)^{\alpha}}\mathrm{d}r\mathbf{1}_{D_{t}^{N,R}}\Big)^2\mathrm{d}t\Big]\cr
    &\leq&C\mathbb{E}\Big[\int_{0}^{T}\big(\Lambda^{0,T}_{\alpha,B^{H,N,n,m}}\big)^2(1+\|u^{N,n}\|_{\alpha,t}^2)\mathbf{1}_{D_{t}^{N,R}}\mathrm{d}t\Big]\cr
    &\leq& C_{N,R, T, \alpha,\beta}\mathbb{E}\big[\big(\Lambda^{0,T}_{\alpha,B^{H,N,n,m}}\big)^2\big],\end{aligned}$$ where $\Lambda^{0,T}_{\alpha,B^{H,N,n,m}}:=\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}\|\beta_{i}^{H,N,n}-\beta_{i}^{H,N,m }\|_{\alpha,0,T}.$

Next, for $\mathcal{N}_{33},$ and $\mathcal{N}_{34}$, by Lemma \[itofbm2\], we have $$\begin{aligned}
&&\mathcal{N}_{33}+\mathcal{N}_{34}\cr
    &\leq&C \mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t}\frac{\big|\int_{s}^{t} S_{t-r} (g(u^{N,n}_r)-g(u^{N,m}_r)) \mathrm{d}B^{H,N,m}_r\big|}{(t-s)^{1+\alpha}}\mathrm{d}s\mathbf{1}_{D_{t}^{N,R}}\Big)^2\mathrm{d}t\Big]\cr
    &&+C \mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t}\frac{\big| \int_{0}^{s} (S_{t-r}-S_{s-r}) (g(u^{N,n}_r)-g(u^{N,m}_r)) \mathrm{d}B^{H,N,m}_r\big|}{(t-s)^{1+\alpha}}\mathrm{d}s\mathbf{1}_{D_{t}^{N,R}}\Big)^2\mathrm{d}t\Big]\cr
    &\leq& C_{N}\mathbb{E}\Big[\int_{0}^{T} \int_{0}^t (t-r)^{-\alpha}[ r^{-\alpha}+(t-r)^{-\alpha}]|u^{N,n}_r-u^{N,m}_r|^2\mathrm{d}r\mathbf{1}_{D_{t}^{N,R}}\mathrm{d}t\Big]\cr
    &&+C_{N}\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t \frac{\int_{0}^{r}|u^{N,n}_r-u^{N,m}_r-u^{N,n}_q+u^{N,m}_q|{(r-q)^{-1-\alpha}}\mathrm{d}q}{(t-r)^\alpha}\mathrm{d}r \mathbf{1}_{D_{t}^{N,R}} \Big)^2 \mathrm{d}t\Big]\cr
    &&+C_{N}\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t \frac{|u^{N,n}_r-u^{N,m}_r|}{ (t-r)^{\alpha}}\Big(\int_{0}^{r}\frac{|u^{N,n}_r-u^{N,n}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r\mathbf{1}_{D_{t}^{N,R}}\Big)^2\mathrm{d}t\Big]\cr
    &&+C_{N}\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^t \frac{ |u^{N,n}_r-u^{N,m}_r|}{ (t-r)^{\alpha}}\Big(\int_{0}^{r}\frac{|u^{N,m}_r-u^{N,m}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r\mathbf{1}_{D_{t}^{N,R}}\Big)^2\mathrm{d}t\Big]\cr
    &\leq& C_{N}\mathbb{E}\Big[\int_{0}^{T} \int_{0}^t (t-r)^{-\alpha}[ r^{-\alpha}+(t-r)^{-\alpha}]|u^{N,n}_r-u^{N,m}_r|^2\mathrm{d}r\mathbf{1}_{D_{t}^{N,R}}\mathrm{d}t\Big]\cr
    &&+C_N\mathbb{E}\Big[\int_{0}^{T}\int_{0}^t \Big(\int_{0}^{r}\frac{|u^{N,n}_r-u^{N,m}_r-u^{N,n}_q+u^{N,m}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)^2\mathrm{d}r\mathbf{1}_{D_{t}^{N,R}}\mathrm{d}t\Big]\cr
    &&+C_{N}\mathbb{E}\Big[\int_{0}^{T}\int_{0}^t |u^{N,n}_r-u^{N,m}_r|^2\Big(\int_{0}^{r}\frac{|u^{N,n}_r-u^{N,n}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)^2\mathrm{d}r\mathbf{1}_{D_{t}^{N,R}}\mathrm{d}t\Big]\cr
    &&+C_{N}\mathbb{E}\Big[\int_{0}^{T}\int_{0}^t |u^{N,n}_r-u^{N,m}_r|^2\Big(\int_{0}^{r}\frac{|u^{N,m}_r-u^{N,m}_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)^2\mathrm{d}r\mathbf{1}_{D_{t}^{N,R}}\mathrm{d}t\Big]\cr
    &\leq& C_{N,R}\int_{0}^{T} \mathbb{E}\big[\|u^{N,n}-u^{N,m}\|^2_{\alpha,t}\mathbf{1}_{D_{t}^{N,R}}\big]\mathrm{d}t.\end{aligned}$$

