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Semi-analytical solution of a McKean–Vlasov equation with feedback through hitting a boundary

Part of: Representations of solutions Integral equations, integral transforms Parabolic equations and systems Stochastic processes

Published online by Cambridge University Press:  16 December 2019

ALEXANDER LIPTON ,
VADIM KAUSHANSKY  and
CHRISTOPH REISINGER Open the ORCID record for CHRISTOPH REISINGER [Opens in a new window]
ALEXANDER LIPTON
Affiliation:
Massachusetts Institute of Technology, Connection Science, Cambridge, MA, USA Silamoney, Portland, OR, USA, email: alexlipt@mit.edu
VADIM KAUSHANSKY
Affiliation:
Mathematical Institute, Oxford-Man Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, UK, emails: vadim.kaushansky@maths.ox.ac.uk; christoph.reisinger@maths.ox.ac.uk
CHRISTOPH REISINGER
Affiliation:
Mathematical Institute, Oxford-Man Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, UK, emails: vadim.kaushansky@maths.ox.ac.uk; christoph.reisinger@maths.ox.ac.uk
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Abstract

In this paper, we study the nonlinear diffusion equation associated with a particle system where the common drift depends on the rate of absorption of particles at a boundary. We provide an interpretation of this equation, which is also related to the supercooled Stefan problem, as a structural credit risk model with default contagion in a large interconnected banking system. Using the method of heat potentials, we derive a coupled system of Volterra integral equations for the transition density and for the loss through absorption. An approximation by expansion is given for a small interaction parameter. We also present a numerical solution algorithm and conduct computational tests.

Keywords

McKean–Vlasov equationssupercooled Stefan problemVolterra equationsstructural credit contagion modelsemi-analytic solution

MSC classification

Primary: 35K58: Semilinear parabolic equations 65R20: Integral equations
Secondary: 35C20: Asymptotic expansions 60G99: None of the above, but in this section
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 6: Special issue featuring papers on Professor Sam Howison , December 2021 , pp. 1035 - 1068
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Copyright
© The Author(s), 2019. Published by Cambridge University Press

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Footnotes

†

Vadim Kaushansky gratefully acknowledges support from the Economic and Social Research Council and Bank of America Merrill Lynch.

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