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

Abstract

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${}_{\mathrm{loc}}$ 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.

28 pages