Finance and Stochastics

, Volume 23, Issue 3, pp 535–594 | Cite as

An SPDE model for systemic risk with endogenous contagion

  • Ben Hambly
  • Andreas SøjmarkEmail author


We propose a dynamic mean-field model for ‘systemic risk’ in large financial systems, derived from a system of interacting diffusions on the positive half-line with an absorbing boundary at the origin. These diffusions represent the distances-to-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 causes 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 characterised 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. Depending on the realisations 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.


Systemic risk Contagion Common noise Mean-field type SPDE on half-line Conditional McKean–Vlasov problem Particle system 

Mathematics Subject Classification (2010)

60H15 60F17 82C22 91G40 91G80 

JEL Classification

G01 G21 G32 



We thank two anonymous referees and the Associate Editor for their careful reading and suggestions for improvements, which has helped sharpen the presentation of this paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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