relations $\left\|f'\right\|_{\theta;t}^{*p}\leq(\left\|f'\right\|_{\infty,\infty;t})^{p}$, $\left\||g'|^2\right\|_{\theta;t}^{*p/2}\leq(\left\||g'|^2\right\|_{\infty,\infty;t})^{p/2}$ and $\left\||h'|^2\right\|_{\theta;t}^{*p/2}\leq(\left\||h'|^2\right\|_{\infty,\infty;t})^{p/2}$ and assumption [**(HO$\mathbf{\infty p}$))**]{} yield $$E\left(\left\|f'\right\|_{\theta;t}^{*p}+\left\||g'|^2\right\|_{\theta;t}^{*p/2}+\left\||h'|^2\right\|_{\theta;t}^{*p/2}\right)<+\infty.$$

As the proof of this theorem is quite long, we split it into several steps.

The case where $\xi$, $\bar{f}^0$, $\bar{g}^0$ and $\bar{h}^0$ are uniformly bounded {#casborn}
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In this subsection, we assume that the hypotheses [**(H)**]{}, [**(O)**]{}, [**(HI$\mathbf{2
p}$)**]{}, [**(HO$\mathbf{\infty p}$)**]{} hold and we add the following stronger ones: $$\label{stronger1}\xi\in
L^\infty(\Omega\times{\ensuremath{\mathcal O}}),$$and $$\bar{f}^0,\ \bar{g}^0,\ \bar{h}^0\in L^\infty(\mathbb{R}_+\times\Omega\times{\ensuremath{\mathcal O}}).$$ Then it is obviously that $\xi-S'_0\in L^\infty(\Omega\times{\ensuremath{\mathcal O}})$.\
Under these hypotheses, we know that the SPDE with obstacle (\[SPDEO\]) admits a unique weak solution $(u,\nu) = \mathcal R
(\xi, f,g,h,S)$ and that $(u-S',\nu)={\ensuremath{\mathcal R}}(\xi-S'_0,\bar{f},\bar{g},\bar{h},S-S')$. We start by proving the following $L^l-$estimate:

\[estimateul\] The solution $u$ of the problem (\[SPDEO\]) belongs to $\cap_{l\geq2}L^l([0,T]\times{\ensuremath{\mathcal O}}\times\Omega)$. Moreover there exist constants $c,\ c'>0$ which only depend on $\ C,\ \alpha,\ \beta$ and on the quantity $$\begin{aligned}
K=\left\|\xi-S'_0\right\|_{L^\infty(\Omega\times{\ensuremath{\mathcal O}})}\vee\left\|\bar{f}^0\right\|_{L^\infty(\mathbb{R}_+\times\Omega\times{\ensuremath{\mathcal O}})}\vee\left\|\bar{g}^0\right\|_{L^\infty(\mathbb{R}_+\times\Omega\times{\ensuremath{\mathcal O}})}\vee\left\|\bar{h}^0\right\|_{L^\infty(\mathbb{R}_+\times\Omega\times{\ensuremath{\mathcal O}})}\end{aligned}$$ such that, for all real $l\geq2$, $$\label{uS'l}E\int_{\ensuremath{\mathcal O}}|u_t(x)-S'_t(x)|^ldx\leq cK^2l(l-1)e^{cl(l-1)t}$$ $$\label{nablauS'l}E\int_0^t\int_{\ensuremath{\mathcal O}}|u_s(x)-S'_s(x)|^{l-2}|\nabla(u_s(x)-S'_s(x))|^2dxds\leq c'K^2l(l-1)e^{cl(l-1)t}$$ and$$\label{nuS'l}E\int_0^t\int_{\ensuremath{\mathcal O}}|u_s(x)-S'_s(x)|^{l-1}\nu(dxds)<+\infty.$$

Notice first that if $(u-S',\nu)={\ensuremath{\mathcal R}}(\xi-S'_0,\bar{f},\bar{g},\bar{h},S-S')$, then $$\bar{f}\left(u-S',\nabla(u-S')\right)
,\bar{g}_i\left(u-S',\nabla(u-S')\right) ,\bar{h}_i\left(
u-S',\nabla(u-S')\right) \in L^2\left([0,T];L^2\left( \Omega \times
{\cal O}\right) \right)$$ and consequently we can apply Itô’s formula to $(u-S',\nu)$ (See Theorem 5 in [@DMZ12]).\
We fix a real $l\geq2$, $T > 0$ 