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2021 At the mercy of the common noise: blow-ups in a conditional McKean–Vlasov Problem
Sean Ledger, Andreas Søjmark
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Electron. J. Probab. 26: 1-39 (2021). DOI: 10.1214/21-EJP597

Abstract

We extend a model of positive feedback and contagion in large mean-field systems, by introducing a common source of noise driven by Brownian motion. Although the driving dynamics are continuous, the positive feedback effect can lead to ‘blow-up’ phenomena whereby solutions develop jump-discontinuities. Our main results are twofold and concern the conditional McKean–Vlasov formulation of the model. First and foremost, we show that there are global solutions to this McKean–Vlasov problem, which can be realised as limit points of a motivating particle system with common noise. Furthermore, we derive results on the occurrence of blow-ups, thereby showing how these events can be triggered or prevented by the pathwise realisations of the common noise.

Funding Statement

This project was initiated while A. Søjmark was at the University of Oxford, supported by the EPSRC award EP/L015811/1.

Acknowledgments

We thank Ben Hambly for initial discussions and advice at the onset of this project. Moreover, we thank two anonymous referees and the associate editor for their encouragement to expand on certain results in Sections 2 and 3 as well as their suggestions for improvement of the overall presentation.

Citation

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Sean Ledger. Andreas Søjmark. "At the mercy of the common noise: blow-ups in a conditional McKean–Vlasov Problem." Electron. J. Probab. 26 1 - 39, 2021. https://doi.org/10.1214/21-EJP597

Information

Received: 1 August 2018; Accepted: 8 February 2021; Published: 2021
First available in Project Euclid: 23 March 2021

Digital Object Identifier: 10.1214/21-EJP597

Subjects:
Primary: 34B16, 60G57, 60H15, 60H30, 82C22

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