O}}\varphi'_n(u_s-S'_s)\nu(dxds)=-\int_0^t\int_{\ensuremath{\mathcal O}}\varphi'_n(S_s-S'_s)\nu(dxds)\leq C.$$ This yields (\[nuS’l\]) by Fatou’s lemma.

With the help of Lemma \[estimateul\], we are able to prove the following Itô formula:

\[Lp\] Assume the hypotheses of the previous lemma. Let $(u,\nu)$ be the solution of the problem (\[SPDEO\]). Then for $
l\geq2$, we get the following Itô’s formula, $ P$-almost surely, for all $t\in [0,T]$, $$\label{LpIt\^o's}
\begin{split}
&  \int_{\mathcal{O}}\left| u_t (x)-S'_t(x) \right|^l  \, dx  +
\int_0^t {\ensuremath{\mathcal E}}\, \big(l\, (u_s-S'_s)^{l -1}sgn(u_s-S'_s) , \,
u_s-S'_s \big) \, ds =
\int_{\mathcal{O}} \left|\xi (x)-S'_0(x) \right|^l \, dx  \\
& +l\int_{0}^{t}\int_{\mathcal{O}}sgn(u_s-S'_s)
 \left|u_s (x)-S'_s(x) \right|^{l-1} \bar{f} (s,x, u_s-S'_s,\nabla (u_s-S'_s)) \, dx ds\\
 & -l (l-1) \, \sum_{i=1}^{d} \,\int_{0}^{t} \int_{\mathcal{O}} \left|u_s
(x)-S'_s(x)  \right|^{l -2}
\partial_i (u_{s}(x)-S'_s(x))\, \bar{g}_{i} (s, x,u_s-S'_s,\nabla (u_s-S'_s)) \, dx \, ds \\
& + l \, \sum_{j=1}^{\infty} \int_{0}^{t}
\int_{\mathcal{O}}sgn(u_s-S'_s)
 \,  \left|u_s (x)-S'_s(x) \right|^{l-1}\bar{h}_{j}( s,x, u_s-S'_s,\nabla (u_s-S'_s))\, dx dB_{s}^{j}\\
 & + \frac{l (l-1)}{2}\, \sum_{j=1}^{\infty}
\int_{0}^{t}\int_{\mathcal{O}}
 \left|u_s (x)-S'_s(x) \right|^{l-2} \bar{h}_{j}^2(s,x, u_s-S'_s,\nabla (u_s-S'_s)) \, dx\, ds\\&+l\int_{0}^{t}\int_{\mathcal{O}}sgn(u_s-S'_s)
 \left|u_s (x)-S'_s(x) \right|^{l-1} \nu(dx\, ds)\, .\\
\end{split}$$

From Itô’s formula (see Theorem 5 in [@DMZ12]), with the same notations as in the previous lemma, we have $ P$-almost surely, and for all $t\in [0,T]$ and all $ n \in {{\ensuremath{\mathbb N}} }^*$, $$\begin{split}
& \int_{\mathcal{O}} \varphi_n (u_t 