Supremum of a Brownian semi-martingale

Question

Consider the process $p_t$ with $t\in[0,1]$ $$ p_t = p_0 +\int_0^t\mu_s\,ds+\int_0^t\nu_s\,dW_s $$ where $\mu$ and $\nu$ are bounded predictable processes and $W$ is a Brownian motion. Let's assume that $p_t\in(0,1)$ (it is always possible to create such a Brownian semi-martingale considering any continuous and differentiable $F:\mathbb{R}\rightarrow (0,1)$ and a Brownian semi-martingale $X$, then taking $p_t=F(X_t)$, by the Ito-lemma, we have that $p_t$ is still a Brownian semi-martingale). Since the trajectories of $p$ are continuous and $[0,t]$ is a compact set we can say that the maximum of $p_s$ over the interval $[0,t]$, call it $\overline{p}_t$, exists and it is strictly contained in $(0,1)$, that is $$ \exists\overline{p}_t\doteq \max_{0\leq s\leq t}p_s\in (0,1) $$ where $\doteq$ means "definition". My problem is: how much restrictive is to assume that $$ \widetilde{p}_t \doteq \sup_{\omega\in\Omega}\overline{p}_t(\omega)\in(0,1)\quad ? $$ Or, better, am I allowed to put such a restriction?