This is a pretty straightforward question for which I have not yet managed to find an answer on the internet. In Wikipedia's article on the Optional Sampling Theorem (OST), the discrete version of the theorem is stated; one of the possible conditions listed for the theorem to hold is:
(c) There exists a constant $c$ such that $|X_{t \wedge \tau}| \leq c$ a.s. for all $t \in \mathbb{N}_0$ where $\wedge$ denotes the minimum operator.
where $X_t$ is the martingale and $\tau$ the stopping time.
Does this condition also hold for the continuous version of the theorem: $t \in \mathbb{R}_+$?
Any reference would be appreciated.
Yes, and the proof is the same: since $t\mapsto X_{t\wedge\tau}$ is a bounded martingale, it converges almost surely and in $L^1$ to some random variable which we call $X_\tau$ (note that $X_\tau(\omega)=X_{\tau(\omega)}(\omega)$ for all $\omega\in\{\tau<\infty\}$, so the notation is justified). By the martingale property, we have $$E[X_\tau]=\lim_{t\to\infty}\mathbb E[X_{t\wedge\tau}]=\lim_{t\to\infty}\mathbb E[X_0]=\mathbb E[X_0].$$