At the Mercy of the Common Noise: Blow-ups in a Conditional McKean--Vlasov Problem

Sean Ledger; Andreas Sojmark

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arXiv · 1807.05126

At the Mercy of the Common Noise: Blow-ups in a Conditional McKean--Vlasov Problem

Sean Ledger·Andreas Sojmark

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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.

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BibTeX RIS

Sean Ledger, Andreas Sojmark. 2018-07-13. At the Mercy of the Common Noise: Blow-ups in a Conditional McKean--Vlasov Problem. https://arxiv.org/abs/1807.05126

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