$\omega\in\Omega$, the map $(t,x)\rightarrow S_t (\omega,x)$ is quasi-continuous. Moreover, $S_0 \leq \xi$ $P$-almost surely and $S$ is controlled by the solution of an SPDE, i.e. $\forall
t\in[0,T],$ $$S_t\leq S'_t,\quad dP\otimes dt\otimes dx-a.e.$$ where $S'$ is the solution of the linear SPDE $$\left\{\begin{array}{ccl} \label{obstacle}
 dS'_t&=&LS'_tdt+f'_tdt+\sum_{i=1}^d \partial_i g'_{i,t}dt+\sum_{j=1}^{+\infty}h'_{j,t}dB^j_t\\
                  S'(0)&=&S'_0 ,
\end{array}\right.$$ with null boundary Dirichlet conditions.\
**Assumption (OL):** The obstacle $S: [0,T]\times \Omega\times
{\ensuremath{\mathcal O}}\rightarrow {\mathbb{R}}$ is an adapted random field, almost surely quasi-continuous, such that $S_0 \leq \xi$ $P$-almost surely and controlled by a **local** solution of an SPDE, i.e. $\forall
t\in[0,T],$ $$S_t\leq S'_t,\quad dP\otimes dt\otimes dx-a.e.$$ where $S'$ is a **local** solution of the linear SPDE $$\left\{\begin{array}{ccl}
 dS'_t&=&LS'_tdt+f'_tdt+\sum_{i=1}^d \partial_i g'_{i,t}dt+\sum_{j=1}^{+\infty}h'_{j,t}dB^j_t\\
                  S'(0)&=&S'_0 .
\end{array}\right.$$ **Assumption (HO2)** $$E \left(\|\xi \|_2^2   +\left\| f'\right\|_{2,2;T}^2+\left\|
|g'|\right\| _{2,2;T}^2+\left\| |h'|\right\| _{2,2;T}^2\right)
<\infty .$$\
**Assumption (HOL)** $$E \int_K |S'_0|^2 dx + E \, \int_{0}^T
\int_K \big( |f'_t(x)|^2 + |g'_t (x)|^2 + |h'_t (x)|^2 \, \big) dx
dt < \infty$$for any compact set $K \subset \mathcal{O}$.

\[remark3\] It is well-known that under [**(HO2)**]{} $S'$ belongs to ${\ensuremath{\mathcal H}}_T$, is unique and satisfies the following estimate: $$\label{estimobstacle1}
E\sup_{t\in[0,T]}\parallel
S'_t\parallel^2+E\int_0^T\mathcal{E}(S'_t)dt\leq CE\left[\parallel
S'_0\parallel^2+\int_0^T(\parallel f'_t\parallel^2+\parallel
|g'_t|\parallel^2+\parallel |h'_t|\parallel^2)dt\right],$$ see for example Theorem 8 in [@DenisStoica]. Moreover, as a consequence of Theorem 3 in [@DMZ12], 