Given a bounded sequence $\{ X^{ n}\}_{ n}$ of semimartingales on a time interval $[0,T]$, we find a sequence of convex combinations $\{ Y^{ n}\}_{ n}$ and a limiting semimartingale $Y$ such that $\{ Y^{ n}\}_{ n}$ converges to $Y$ in a $\sigma$-localized modification of the \'Emery topology. More precisely, $\{ Y^{ n}\}_{ n}$ converges to $Y$ in the \'Emery topology on an increasing sequence $\{ D_{ n}\}_{ n}$ of predictable sets covering $\Omega\times[0,T]$. We also prove some technical variants of this theorem, including a version where the complement of $\{ D_{ n}\}_{ n}$ forms a disjoint sequence. Applications include a complete characterization of sequences admitting convex combinations converging in the \'Emery topology, and a supermartingale counterpart of Helly's selection theorem.
Categories: math.PR math.FA