At the mercy of the common noise: blow-ups in a conditional McKean–Vlasov Problem

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.

Optimal bailout strategies resulting from the drift controlled supercooled Stefan problem

“…2 We consider L T − rather than L T in the constraint because Λ may have a jump discontinuity precisely at T , which would considerably complicate the analysis. However, by [39,Corollary 2.3], we know that solutions to (1) cannot have discontinuities after time α 2 /(2π). We can derive an analogous result for the controlled case using Girsanov's theorem if, in addition to the pointwise bound on β, we assume a bound on the total cost over the infinite horizon, i.e.…”

“…Indeed, it is known that the McKean-Vlasov problem (2) and (3) may admit more than one solution, and it is not known that physical solutions exist for general β ∈ B T , although it is known for β of the special form b(t, X t ), where b is Lipschitz (see e.g. [26,39]). To pin down a meaningful solution concept, we therefore rely on the notion of minimal solutions as defined in (5).…”

“…We now explain our numerical procedure to solve (52). As in Section 3.2 we work with the equivalent PDE (39) and use the iterative procedure (41). We start with the uncontrolled system, i.e.…”

Dynamic Default Contagion in Heterogeneous Interbank Systems

“…Moreover, we are interested in making a precise connection between this dynamic balance sheet framework and the recent probability literature on mean-field approaches to contagion modelling [18,19,21,22,23]. This, of course, also links to the financial mathematics literature on mean-field models for systemic risk, as exemplified by [5,7,13], although these works have focused on 'flocking effects' modelled by mean-reversion rather than default contagion.…”

McKean-Vlasov Equations with Positive Feedback through Elastic Stopping Times

“…McKean-Vlasov equations with positive feedback through hitting times have recently been used in the modelling of a range of phenomena, from the self-excitation of neurons [4], [5], through the modelling of default and systemic risk in banking networks, [6], [7], [8], [9], [10], to a probabilistic representation for the supercooled Stefan problem [3]. In these models, the feedback is generated through the dependence of the process on the distribution of its hitting times.…”