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.
Highlights
This paper studies a model of positive feedback and contagion in large mean-field systems, focusing on the interplay between positive feedback loops and a common source of noise in the driving dynamics
The common noise, B0, plays a pivotal role: in certain cases it has the power to provoke or prevent a blow-up, where a blow-up is defined as a jump discontinuity of t → Lt
A key property of (MV) is that these blow-ups occur endogenously: all the random drivers are continuous, yet jump discontinuities can develop through the positive feedback effect alone
Summary
This paper studies a model of positive feedback and contagion in large mean-field systems, focusing on the interplay between positive feedback loops and a common source of noise in the driving dynamics. We show that solutions to (MV) arise as large population limits of a ‘contagious’ particle system, which provides the principal theoretical justification for the various applications discussed below We establish this convergence result for more general drift and diffusion coefficients than the case (MV), and we allow for a large class of natural initial conditions (see Theorem 3.2). To get a better feel for the workings of (MV), consider the conditional law of X, with absorption at the origin, given the common noise B0 This defines a flow of random sub-probability measures t → νt, where νt := P(Xt ∈ · , t < τ | B0), for all t ≥ 0, describing how the ‘surviving’ mass of the system evolves.
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