Let $X_t$ be an american style financial derivative with random exercise time $T$
where $t$ and $T$ belongs to some finite set $A$.

Buying this derivative requires the buyer to pay $p_t$ up to time $T$.

Let $\Omega$ be the sample space of $X_t$, $p=(p_t)_{t \in A}$ the price process
and $B={\left(C^A \right)}^\Omega$ the value space of $p$ for some set $C \subset \mathbb{R}$.

Assume expectation are taken under the risk-neutral measure with $B_t$ as the risk-free discounting factor from times $0$ to time $t$.

Is the no-arbitrage pricing process of the derivative given by
$$
\arg_{p \in B} \left( \sup_{T \in A} E(X_T B_T - \int_0^T p_t B_tdt)=0 \right) \text{ (1)}
$$

when $B$ requires that $p_t(\omega) \ne 0, t>0$ for some $\omega \in \Omega$?

My knowledge of finance tells me the no-arbitrage price would be
$$
\sup_{T \in A} E(X_T B_T) \text{ (2)}
$$
when B is degenerated to $p_0=k$ for some $k \in \mathbb{R}$ and $p_t=0,t \ne 0$.

Intuitively, I would expect (1) to be the natural extension to (2).

But is it theoretically true?

I searched, but I couldn't find any source confirming my hypothesis.

Thanks for any help.