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

TL;DR

This paper establishes limit theorems for $\sigma$-localized Emery convergence of semimartingales, providing new convergence results and applications in stochastic analysis.

Contribution

It introduces new limit theorems for $\sigma$-localized Emery convergence, including convex combination convergence and a supermartingale Helly's theorem variant.

Findings

Convex combinations of semimartingales converge in the $\sigma$-localized Emery topology.

Characterization of sequences that admit such convex combination convergence.

A supermartingale version of Helly's selection theorem is proved.

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.