This might be a fairly silly question, but here it goes. I'm initiating my study of optimal stopping problems (OSP), and I was wondering if an OSP can be seen as a function in some $L^p$ space. Specifically, let us consider an underlying probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a stochastic process $X = (X_t)_{t\in [0,T]}$ taking values in some state space $S$, i.e. $X$ can be seen as a map $\Omega \to S$. For the filtration, simply consider the filtration generated by $X$, i.e. $\mathcal{F}_t^X = \sigma(X_s : 0 \leq s \leq t)$.
Now, let us consider an OSP of the form $$\sup_{\tau \in \mathcal{S}} \mathbb{E}[Z_\tau],$$ where $\mathcal{S}$ is the set of $(\mathcal{F}^X_t)$-adapted stopping times in $[0,T]$, and $Z = (Z_t)_{t\in [0,T]}$ is a real-valued stochastic process adapted to $(\mathcal{F}^X_t)_{t\in [0,T]}$. For simplicity let us even assume that $Z_t = \phi(X_t)$, where $\phi: S \to \mathbb{R}$ is some function for which we may impose conditions on. We then get an OSP of the form $$\sup_{\tau \in \mathcal{S}} \mathbb{E}[\phi(X_\tau)].$$
My question is: can we look at the OSP above as a function in some $L^p(S)$, i.e. as a function $$S \ni X \mapsto \sup_{\tau \in \mathcal{S}} \mathbb{E}[\phi(X_\tau)],$$ whose $pth$ power is integrable? Since $L^2(S)$ is a Hilbert space, I'm mostly interested in the case $p=2$. What sort of conditions would one need to impose on $\phi$? Any insights will be appreciated, including references for further reading.