we know that $S'$ admits a quasi-continuous version.

A pair $(u,\nu)$ is said to be a solution of the problem (\[SPDEO\]) if

1.  $u\in\mathcal{H}_T$, $u(t,x)\geq S(t,x),\ dP\otimes dt\otimes
        dx-a.e.$ and $u_0(x)=\xi,\ dP\otimes dx-a.e.$;

2.  $\nu$ is a random regular measure defined on $[0,T[\times\mathcal{O}$;

3.  the following relation holds almost surely, for all $t\in[0,T]$ and all $\varphi\in\mathcal{D}$, $$\begin{split}\label{solution}(u_t,\varphi_t)=&(\xi,\varphi_0)+\int_0^t(u_s,\partial_s\varphi_s)ds-\int_0^t\mathcal{E}(u_s,\varphi_s)ds\\&-\sum_{i=1}^d\int_0^t(g^i_s(u_s,\nabla u_s),\partial_i\varphi_s)ds
        +\int_0^t(f_s(u_s,\nabla u_s),\varphi_s)ds\\&+\sum_{j=1}^{+\infty}\int_0^t(h^j_s(u_s,\nabla u_s),\varphi_s)dB^j_s+\int_0^t\int_{\mathcal{O}}\varphi_s(x)\nu(dx,ds);\end{split}$$

4.  $u$ admits a quasi-continuous version, $\tilde{u}$, and we have $$\int_0^T\int_{\ensuremath{\mathcal O}}(\tilde{u}(s,x)-S(s,x))\nu(dx,ds)=0,\ \
        P-a.s.$$

We denote by ${\ensuremath{\mathcal R}}(\xi,f,g,h,S)$ the solution of the obstacle problem when it exists and is unique.

A pair $(u,\nu)$ is said to be a local solution of the problem (\[SPDEO\]) if

1.  $u\in\mathcal{H}_{loc}$, $u(t,x)\geq S(t,x),\ dP\otimes dt\otimes
        dx-a.e.$ and $u_0(x)=\xi,\ dP\otimes dx-a.e.$;

2.  $\nu$ is a local random regular measure defined on $[0,T[\times\mathcal{O}$;

3.  the following relation holds almost surely, for all $t\in[0,T]$ and all $\varphi\in\mathcal{D}$, $$\begin{split}\label{solutionlocal}(u_t,\varphi_t)=&(\xi,\varphi_0)+\int_0^t(u_s,\partial_s\varphi_s)ds-\int_0^t\mathcal{E}(u_s,\varphi_s)ds\\&-\sum_{i=1}^d\int_0^t(g^i_s(u_s,\nabla u_s),\partial_i\varphi_s)ds
        +\int_0^t(f_s(u_s,\nabla u_s),\varphi_s)ds\\&+\sum_{j=1}^{+\infty}\int_0^t(h^j_s(u_s,\nabla u_s),\varphi_s)dB^j_s+\int_0^t\int_{\mathcal{O}}\varphi_s(x)\nu(dx,ds);\end{split}$$

4.  $u$ admits a quasi-continuous version, $\tilde{u}$, and we have $$\int_0^T\int_{\ensuremath{\mathcal O}}(\tilde{u}(s,x)-S(s,x))\nu(dx,ds)=0,\ \
        P-a.s.$$

We denote by ${\ensuremath{\mathcal R}}_{loc}(\xi,f,g,h,S)$ the set of all the local solutions $(u,\nu)$.\
Finally, in the sequel, we introduce some constants $\epsilon$, $\delta>0$, we shall denote by $C_\epsilon$, $C_\delta$ some constants depending only on $\epsilon$, $\delta$, typically those appearing 