Limit theorems for $\sigma$-localized \'Emery convergence

Abstract

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.