A McKean--Vlasov equation with positive feedback and blow-ups

Hambly, Ben; Ledger, Sean; Søjmark, Andreas

doi:10.48550/arxiv.1801.07703

Cited by 9 publications

(27 citation statements)

References 3 publications

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“…This is useful in applications of Proposition 2.4 below. While we focus on f (x) = x in the present paper, as in the motivating papers [9,10,15,16], we note that our results extend immediately to (2.4) for any globally Lipschitz continuous f . Also, it takes only minor adaptations (see [18,Sec.…”

Section: Transformed Loss Processesmentioning

confidence: 59%

“…Questions of existence and uniqueness for (MV) are delicate, not least because of the possibility of blow-ups. At present, results are only available for the case ρ = 0 [9,10,15,18], with uniqueness only known before the first blow-up [15,18]. In the present paper we establish global existence of a 'relaxed' version of (MV) via limit points of the finite particle system, which provides the justification for the aforementioned mean-field models.…”

Section: (Mv)mentioning

confidence: 64%

“…Proof. If ν 0 is supported on (0, 1 2 α), then EX 0 < 1 2 α, so the first part is immediate from [15,Thm. 1.1].…”

Section: The Idiosyncratic Model With ρ =mentioning

confidence: 99%

“…The above model without the common noise motivates the analysis in [15] and numerical schemes for this model have been explored in [21]. A similar financial framework also provides the motivation for [18,19]-the latest paper extending the idiosyncratic (ρ = 0) model to more generic random drivers (playing the role of the B here) and replacing the constant feedback parameter with a graph of interactions.…”

Section: (Mv)mentioning

confidence: 99%

“…We begin this section by recalling the notion of a physical solution for (MV) with ρ = 0 (see the earlier treatments in [9,15,18]), which stipulates that X and L must be càdlàg with jump sizes satisfying the equation…”

Section: The Possibility and Timing Of Blow-upsmentioning

confidence: 99%

See 4 more Smart Citations

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

Ledger¹,

Søjmark²

2018

Preprint

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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 dynamics in the model are continuous, the feedback effect can lead to jump discontinuities in the solutions, which we refer to as 'blow-ups'. We prove existence of solutions to the corresponding conditional McKean-Vlasov equation, by realising them as suitable limit points of the finite-dimensional particle system, and we show that the pathwise realisation of the common noise can both trigger and prevent blow-ups.

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Section: Transformed Loss Processesmentioning

confidence: 59%

Section: (Mv)mentioning

confidence: 64%

“…Proof. If ν 0 is supported on (0, 1 2 α), then EX 0 < 1 2 α, so the first part is immediate from [15,Thm. 1.1].…”

Section: The Idiosyncratic Model With ρ =mentioning

confidence: 99%

Section: (Mv)mentioning

confidence: 99%

Section: The Possibility and Timing Of Blow-upsmentioning

confidence: 99%

See 3 more Smart Citations

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

Ledger¹,

Søjmark²

2018

Preprint

Self Cite

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An SPDE model for systemic risk with endogenous contagion

Hambly

Søjmark

2019

Finance Stoch

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We propose a dynamic mean field model for 'systemic risk' in large financial systems, which we derive from a system of interacting diffusions on the positive half-line with an absorbing boundary at the origin. These diffusions represent the distancesto-default of financial institutions and absorption at zero corresponds to default. As a way of modelling correlated exposures and herd behaviour, we consider a common source of noise and a form of mean-reversion in the drift. Moreover, we introduce an endogenous contagion mechanism whereby the default of one institution can cause a drop in the distances-to-default of the other institutions. In this way, we aim to capture key 'system-wide' effects on risk. The resulting mean field limit is characterized uniquely by a nonlinear SPDE on the half-line with a Dirichlet boundary condition. The density of this SPDE gives the conditional law of a non-standard 'conditional' McKean-Vlasov diffusion, for which we provide a novel upper Dirichlet heat kernel type estimate that is essential to the proofs. Depending on the realizations of the common noise and the rate of mean reversion, the SPDE can exhibit rapid accelerations in the loss of mass at the boundary. In other words, the contagion mechanism can give rise to periods of significant systemic default clustering.

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Dynamic contagion in a banking system with births and defaults

2019

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We consider a dynamic model of interconnected banks. New banks can emerge, and existing banks can default, creating a birth-and-death setup. Microscopically, banks evolve as independent geometric Brownian motions. Systemic effects are captured through default contagion: as one bank defaults, reserves of other banks are reduced by a random proportion. After examining the long-term stability of this system, we investigate mean-field limits as the number of banks tends to infinity. Our main results concern the measure-valued scaling limit which is governed by a McKean-Vlasov jump-diffusion. The default impact creates a mean-field drift, while the births and defaults introduce jump terms tied to the current distribution of the process. Individual dynamics in the limit is described by the propagation of chaos phenomenon. In certain cases, we explicitly characterize the limiting average reserves.

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A McKean--Vlasov equation with positive feedback and blow-ups

Cited by 9 publications

References 3 publications

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

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

An SPDE model for systemic risk with endogenous contagion

Dynamic contagion in a banking system with births and defaults

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