Limit theorems for $\sigma$-localized Émery 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 Émery topology. More precisely, $\{Y^n{\}}_n$ converges to $Y$ in the Émery 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 Émery topology, and a supermartingale counterpart of Helly's selection theorem.

Revision from reviewer comments