Characterisation of [formula omitted]-boundedness for a general set of processes with no strictly positive element

Abstract

We consider a general set X of adapted nonnegative stochastic processes in infinite continuous time. 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 X — DSV corresponds to an asymptotic L0-boundedness at the first time all processes in X vanish, whereas NUPBRloc states that Xt={Xt:X∈X} is bounded in L0 for each t∈[0,∞). 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∈X, with an additional asymptotic strict positivity property for Y in the case of DSV.