s\Big)^2\mathrm{d}t\Big] \cr
    &\leq& C.\end{aligned}$$

For $\mathcal{M}_2$, we have $$\begin{aligned}
    \mathcal{M}_2
    &\leq&C\mathbb{E}\Big[\sup _{t \in[0, T]} \Big|\int_{0}^{t} S_{t-s} f(u^{N,n}_s) \mathrm{d}s\Big|^2\Big]\cr
    &&+C\mathbb{E}\Big[\int_{0}^{T}\Big(  \int_{0}^{t}\frac{|\int_{0}^{t}S_{t-r} f(u^{N,n}_r) \mathrm{d} r-\int_{0}^{s}S_{s-r} f(u^{N,n}_r)\mathrm{d}r|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathrm{d}t\Big]\cr
    &=:& \mathcal{M}_{21}+\mathcal{M}_{22}.\end{aligned}$$ By Hölder’s inequality and the growth condition, we get $$\begin{aligned}
    \mathcal{M}_{21}
    &\leq& C \mathbb{E} \Big[\int_{0}^{T} \|S_{t-s}\|^2|f(u^{N,n}_s)|^2\mathrm{d}s\Big] \cr
    &\leq& C \int_{0}^{T}(1+\mathbb{E}[|u^{N,n}_s|^2])\mathrm{d}s.\end{aligned}$$ Next, for $\mathcal{M}_{22}$, by Fubini’s theorem, Lemma \[inq-s\] and Hölder’s inequality, we get $$\begin{aligned}
    \mathcal{M}_{22}&\leq&C\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t}\frac{|\int_{s}^{t}S_{t-r} f(u^{N,n}_r) \mathrm{d}r|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathrm{d}t\Big]\cr
    &&+C \mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t}\frac{|\int_{0}^{s}(S_{t-r}-S_{s-r})f(u^{N,n}_r)\mathrm{d}r|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathrm{d}t\Big]\cr
    &\leq&C\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t} \int_{0}^{r}(t-s)^{-1-\alpha}\mathrm{d}s|f(u^{N,n}_r)|\mathrm{d} r\Big)^2\mathrm{d}t\Big]\cr
    &&+ C\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t} \int_{r}^t (t-s)^{-1-\alpha+{\beta}}(s-r)^{-{\beta}}\mathrm{d}s |f(u^{N,n}_r)|\mathrm{d}r\Big)^2\mathrm{d}t\Big]\cr
    &\leq&C \mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t} (t-r)^{-\alpha} |f(u^{N,n}_r)|d r \Big)^2\mathrm{d}t\Big]\cr
    &\leq&  C \int_{0}^{T}(1+\mathbb{E}[|u^{N,n}_s|^2])\mathrm{d}s.\end{aligned}$$

For $\mathcal{M}_3$, we get $$\begin{aligned}
    \mathcal{M}_3
    &\leq&C\mathbb{E}\Big[\sup _{t \in[0, T]}\Big|\int_{0}^{t} S_{t-s} \sigma(u^{N,n}_s) \mathrm{d} W_s\Big|^2\Big]\cr
    &&+C\mathbb{E}\Big[ \int_{0}^{T}\Big(  \int_{0}^{t}\frac{|\int_{0}^{t}S_{t-r} \sigma(u^{N,n}_r)\mathrm{d} W_r-\int_{0}^{s}S_{s-r} \sigma(u^{N,n}_r)\mathrm{d} W_r|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathrm{d}t\Big]\cr
    &=:& \mathcal{M}_{31}+\mathcal{M}_{32},\end{aligned}$$ and then, by Burkholder-Davis-Gundy inequality, $$\begin{aligned}
    \mathcal{M}_{31}
    &\leq& C \mathbb{E} \Big[\int_{0}^{T} \|S_{t-s}\|^2|\sigma(u^{N,n}_s)|^2_{L_2(V)}\mathrm{d}s\Big]\cr
    &\leq& C \int_{0}^{T}(1+\mathbb{E}[|u^{N,n}_s|^2])\mathrm{d}s.\end{aligned}$$ Applying again Hölder’s inequality, Burkholder-Davis-Gundy inequality, Fubini’s theorem, Lemma \[inq-s\] and Lemma \[inq-s\], we have $$\begin{aligned}
    \mathcal{M}_{32}&\leq&C\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t}\frac{|\int_{s}^{t}S_{t-r} \sigma(u^{N,n}_r) \mathrm{d}W_r|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathrm{d}t\Big]\cr
    &&+C\mathbb{E}\Big[\int_{0}^{T}\Big(\int_{0}^{t}\frac{|\int_{0}^{s}(S_{t-r}-S_{s-r})\sigma(u^{N,n}_r)\mathrm{d}W_r|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathrm{d}t\Big]\cr
    &\leq&C\mathbb{E}\Big[\int_{0}^{T}\int_{0}^{t}(t-s)^{-\frac{3}{2}-\alpha}\Big|\int_{s}^{t}S_{t-r} \sigma(u^{N,n}_r) \mathrm{d}W_r\Big|^2\mathrm{d}s\mathrm{d}t\Big]\cr
    &&+C\mathbb{E}\Big[\int_{0}^{T}\int_{0}^{t}(t-s)^{-\frac{3}{2}-\alpha}\Big|\int_{0}^{s}(S_{t-r}-S_{s-r})\sigma(u^{N,n}_r)\mathrm{d}W_r\Big|^2\mathrm{d}s\mathrm{d}t\Big]\cr
    &\leq&C\int_{0}^{T}\int_{0}^{t}(t-s)^{-\frac{3}{2}-\alpha}\int_{s}^{t}  \|S_{t-r}\| \mathbb{E}[|\sigma(u^{N,n}_r)|^2_{L_{2(V)}}]d rds\mathrm{d}t\cr
    &&+C\int_{0}^{T}\int_{0}^{t} \int_{0}^{s}(t-s)^{-\frac{3}{2}-\alpha}\|S_{t-r}-S_{s-r}\|^2 \mathbb{E}[|\sigma(u^{N,n}_r)|^2_{L_{2(V)}}]\mathrm{d}r \mathrm{d}s \mathrm{d}t\cr
    &\leq&C\int_{0}^{T}\int_{0}^{t}\int_{0}^{r}(t-s)^{-\frac{3}{2}-\alpha} \mathrm{d}s (1+\mathbb{E}[|u^{N,n}_r|^2])\mathrm{d} r\mathrm{d}t\cr
    &&+C\int_{0}^{T} \int_{0}^{t}\int_{r}^{t}(t-s)^{-\frac{3}{2}-\alpha+2\beta'} (s-r)^{-2\beta'}\mathrm{d}s(1+\mathbb{E}[|u^{N,n}_r|^2])\mathrm{d}r\mathrm{d}t\cr
    &\leq& C \int_{0}^{T}(1+\mathbb{E}[\|u^{N,n}\|^2_{\alpha, t}])\mathrm{d}t.\end{aligned}$$

To proceed, for $\mathcal{M}_4$, we obtain $$\begin{aligned}
    \mathcal{M}_4
    &\leq& C \mathbb{E}\Big[\int_{0}^{T}\Big(  \int_{0}^{t}\frac{|\int_{0}^{t}S_{t-r} g(u^{N,n}_r)\mathrm{d}B^{H,N,n}_r-\int_{0}^{s}S_{s-r} g(u^{N,n}_r)\mathrm{d}B^{H,N,n}_r|}{(t-s)^{1+\alpha}}\mathrm{d}s\Big)^2\mathrm{d}t\Big]\cr
    &&+C\mathbb{E}\Big[\sup _{t \in[0, T]} \Big|\int_{0}^{t} S_{t-r} g(u^{N,n}_r) \mathrm{d}B^{H,N,n}_r\Big|^2\Big]\cr
    &=:& \mathcal{M}_{41}+\mathcal{M}_{42}.\end{aligned}$$

For $\mathcal{M}_{41}$, by (\[betaNn-Nn\]), Lemma \[itofbm2\] and Hölder’s inequality, taking $\alpha<\alpha'<1-\beta$, we 