and introduce the sequence $({\varphi}_n
)_{n\in{{\mathbb{N}}^{\ast}}}$ of functions such that for all $n\in{{\mathbb{N}}^{\ast}}$: $$\forall x\in{\mathbb{R}},\, {\varphi}_n (x)= \left\{
\begin{array}{ll}
 \mid x \mid^l&\makebox{ if }\mid x\mid\leq n\\
 n^{l-2} \, \big[  \frac{l (l-1)}{2} (|x|-n)^2+ l \, n (|x|-n) + n^2\,\big] &\makebox{ if }\mid x\mid >n
 \end{array}\right.$$ One can easily verify that for fixed $n$, ${\varphi}_n$ is twice differentiable with bounded second derivative, $ \varphi_n^{\prime
\prime} (x) \geq 0$, and as $ n \to \infty $ one has $ \varphi_n (x)\longrightarrow |x|^{l}$, $\varphi_n^{\prime} (x)\longrightarrow lsgn(x)|x|^{l-1}$, $\varphi_n^{\prime \prime} (x)\longrightarrow l (l-1)|x|^{l-2}$. Moreover, the following relations hold, for all $x\in {{\ensuremath{\mathbb R}} }$ and $n\geq l$:

1.  $\mid x{\varphi}_n^{\prime}(x)\mid\leq l{\varphi}_n (x). $

2.  $\mid {\varphi}_n^{\prime}(x)\mid\leq\mid
    x{\varphi}_n^{\prime\prime}(x)\mid$.

3.  $\mid x^2{\varphi}_n^{\prime\prime}(x)\mid \leq l(l-1){\varphi}_n (x)$.

4.  $|{\varphi}_n^{\prime}(x)|\leq l({\varphi}_n (x)+1 ).$

5.  $|{\varphi}_n^{\prime\prime}(x)|\leq l(l-1)({\varphi}_n (x)+1 ).$

Applying Itô’s formula to $\varphi_n(u-S')$, we have $P$-a.s. for all $t\in [0,T]$, $$\label{uS'It\^o's}
\begin{split}
& \int_{\mathcal{O}} \varphi_n (u_t (x)-S'_t(x)) \, dx \; + \;
\int_0^t \mathcal{E} \big(\varphi_n^{\prime} (u_s-S'_s), \, u_s-S'_s
\big) \, ds = \int_{\mathcal{O}} \varphi_n (\xi(x)-S'_0(x) ) \, dx
\\ & + \int_{0}^{t}\int_{\mathcal{O}}
\varphi_n^{\prime}(u_s(x)-S'_s(x)) \bar{f}  (s,x, u_s-S'_s,\nabla (u_s-S'_s)) \, dx ds \\
&  - \sum_{i=1}^{d} \int_{0}^{t} \int_{\mathcal{O}}
\varphi_n^{\prime \prime } (u_s(x)-S'_s(x) )
\partial_i (u_s(x)-S'_s(x)) \, \bar{g}_{i} (s,x,u_s-S'_s, \nabla (u_s-S'_s)) \, dx \, 