Weak convergence of stochastic integrals on Skorokhod space in Skorokhod's J1 and M1 topologies

Andreas Sojmark and Fabrice Wunderlich

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

Abstract: We provide criteria for Itô integration to behave continuously with respect to Skorokhod’s J1 and M1 topologies, when the integrands and integrators converge weaklyor in probability. The results are novel in the M1 setting and unify existing theories in the J1 case. Beyond sufficient criteria, we present an example of uniformly convergent martingale integrators for which the continuity breaks down. Moreover, we show that, for families of local martingales, M1 tightness in fact implies J1 tightness under a mild localised uniform integrability condition. Finally, we apply our results to study scaling limits of models of anomalous diffusion driven by continuous-time random walks. This yields new results on weak M1 and J1 convergence to stochastic integrals against subordinated stable processes. In the case of superdiffusive scaling, an interesting counterexample is obtained.

JEL-codes: C1 (search for similar items in EconPapers)
Pages: 53 pages
Date: 2026-03-23
References: Add references at CitEc
Citations:

Published in Probability Theory and Related Fields, 23, March, 2026. ISSN: 0178-8051

Downloads: (external link)
https://researchonline.lse.ac.uk/id/eprint/137477/ Open access version. (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:ehl:lserod:137477

Access Statistics for this paper

More papers in LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library LSE Library Portugal Street London, WC2A 2HD, U.K.. Contact information at EDIRC.
Bibliographic data for series maintained by LSERO Manager ().