# Relative compactness of It\^o integrals on the M1 Skorokhod space with identification of limits

## TL;DR

This paper develops general criteria for the weak relative compactness of Itô integrals in the M1 Skorokhod topology and characterizes their limit points, addressing a significant gap in stochastic process convergence theory.

## Contribution

It introduces new results for weak relative compactness of Itô integrals in the M1 topology and explicitly characterizes limit points, expanding the understanding beyond previous J1 and S topology results.

## Key findings

- Established weak relative compactness criteria in M1 topology
- Explicitly characterized the limit points of convergent subsequences
- Closed a gap in the literature on Skorokhod space topologies

## Abstract

We establish general results for weak relative compactness of sequences of It\^o integrals with respect to Skorohod's functional M1 topology, under general conditions. Moreover, we are able to explicitly characterise the form of the limit points of all convergent subsequences. This result closes a longstanding gap in the literature, where weak relative compactness had previously only been shown under the significantly more restrictive J1 topology, or the S topology which is too coarse to preserve continuity for most common functionals.