I am currently reading through Protter's 'stochastic integration and differential equations' and Bass' 'stochastic processes' books.
Both seem to make use of "bounded" stochastic processes. I am unsure of whether this means
1) there exists $K > 0$ such that for each $(t, \omega) \in [0,\infty) \times \Omega$, $|X(t,\omega)| < K$
2) for each $\omega \in \Omega$, there exists $K > 0$ such that for each $t \in [0,\infty)$, $|X(t, \omega)| <K$.
I could not find the definition in either of these books, so I could be missing it, or it could be assumed to be known? If you know a page number then that would be great.
I suspect that definition 1) is the case, but I just wanted to make sure because I want to show that "a bounded increasing cadlag process is a submartingale" (which I think should be adapted as well?) (Bass pg 130) Bass then goes on to use the Doob Meyer Decomposition on this process, but I think that it should be of class D. I think it would be easier to show it is of class D if the definition of "bounded" was that of 1).