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

Abstract

We establish general results for weak relative compactness of sequences of It\^o integrals with respect to Skorohod's functional M1 topology, under sharp 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.