Characterisation of $L^0$-boundedness for a general set of processes with no strictly positive element

We consider a general set $\mathcal{X}$ of adapted nonnegative stochastic processes in infinite continuous time. $\mathcal{X}$ is assumed to satisfy mild convexity conditions, but in contrast to earlier papers need not contain a strictly positive process. We introduce two boundedness conditions on $\mathcal{X}$ -- DSV corresponds to an asymptotic $L^0$-boundedness at the first time all processes in $\mathcal{X}$ vanish, whereas NUPBR$_{\rm loc}$ states that $\mathcal{X}_t = \{ X_t : X \in \mathcal{X}\}$ is bounded in $L^0$ for each $t \in [0,\infty)$. We show that both conditions are equivalent to the existence of a strictly positive adapted process $Y$ such that $XY$ is a supermartingale for all $X \in \mathcal{X}$, with an additional asymptotic strict positivity property for $Y$ in the case of DSV.