Lipschitz conditions with the same Lipschitz coefficients as $f$, $g$ and $h$ and $\left\|\xi-S'_0\right\|_\infty\in
L^p(\Omega,P)$. Nevertheless, we need a supplementary hypothesis:\
**Assumption (HD$\mathbf{\theta p}$)** $$E((\left\|
\bar{f}^0\right\|^*_{\theta;T})^p+(\left\|  |\bar{g}^0
|^2\right\|^*_{\theta;T})^{p/2}+(\left\|  |\bar{h}^0
|^2\right\|^*_{\theta;T})^{p/2})<\infty .$$

This assumption is fulfilled in the following case:

If $\left\|\nabla S'\right\|^*_{\theta;T},\
\left\|f^0\right\|^*_{\theta;T},\ \left\|g^0\right\|^*_{\theta;T}\
and\ \left\|h^0\right\|^*_{\theta;T}$ belong to $ L^p(\Omega,P), $ and assumptions [**(H)**]{} and [**(HO$\mathbf{\infty p}$)**]{} hold, then:\
$\bar{f}$ satisfies the Lipschitz condition with the same Lipschitz coefficients: $$\begin{aligned}
\left|\bar{f}(t,\omega,x,y,z)-\bar{f}(t,\omega,x,y',z')\right|&=&\big|f(t,\omega,x,y+S'_t(x),z+\nabla S'_t(x))+f'(t,\omega,x)\\&-&f(t,\omega,x,y'+S'_t(x),z'+\nabla S'_t(x))-f'(t,\omega,x)\big|\\&\leq& C\left|y-y'\right|+C\left|z-z'\right|.\end{aligned}$$

$\bar{f}$ satisfies the integrability condition: $$\begin{aligned}
\left\|\bar{f}^0\right\|^*_{\theta;T}&=&\left\|f(S',\nabla
S')-f'\right\|^*_{\theta;T}\leq \left\|f(S',\nabla
S')\right\|^*_{\theta;T}+\left\|f'\right\|^*_{\theta;T}\\&\leq&\left\|f^0\right\|^*_{\theta;T}+C\left\|S'\right\|^*_{\theta;T}+C\left\|\nabla
S'\right\|^*_{\theta;T}+\left\|f'\right\|_{\infty,\infty;T}.\end{aligned}$$

And the same for $\bar{g}$ and $\bar{h}$, which proves that [**(HD$\mathbf{\theta p}$)**]{} holds.

We now give the main result of this Section, which is a version of the maximum principle in the case of a solution vanishing on the boundary of ${\ensuremath{\mathcal O}}$:

\[LPESTIM\] Suppose that assumptions [**(H)**]{}, [**(O)**]{}, [**(HI$\mathbf{2
p}$)**]{}, [**(HO$\mathbf{\infty p}$)**]{} and [**(HD$\mathbf{\theta
p}$)**]{} hold, for some $\theta\in[0,1[$ and $p\geq2$ and that the constants of Lipschitz conditions satisfy $$\alpha+\frac{\beta^2}{2}+72\beta^2<\lambda.$$ Let $(u,\ \nu)$ be the solution of OSPDE (\[SPDEO\]) with null boundary condition, then for all $t\in [0,T]$, $$\begin{split}E\left\| u\right\|^p_{\infty,\infty;t}&\leq c(p)k(t)E\big(\left\|\xi\right\|_\infty^p+\left\|S'_0\right\|_\infty^p+\left\|f'\right\|_{\theta;t}^{*p}+\left\||g'|^2\right\|_{\theta;t}^{*p/2}+\left\||h'|^2\right\|_{\theta;t}^{*p/2}\\&\quad+\left\|\bar{f}^0\right\|_{\theta;t}^{*p}+\left\||\bar{g}^0|^2\right\|_{\theta;t}^{*p/2}+\left\||\bar{h}^0|^2\right\|_{\theta;t}^{*p/2}\big),\end{split}$$ where $c(p)$ is a constant which depends on $p$ and $k(t)$ is a constant which depends on the structure constants and $t\in[0,T]$.

The 