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

Published online by Cambridge University Press:  16 December 2019

ALEXANDER LIPTON
Affiliation:
Massachusetts Institute of Technology, Connection Science, Cambridge, MA, USA Silamoney, Portland, OR, USA, email: [email protected]
VADIM KAUSHANSKY
Affiliation:
Mathematical Institute, Oxford-Man Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, UK, emails: [email protected]; [email protected]
CHRISTOPH REISINGER
Affiliation:
Mathematical Institute, Oxford-Man Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, UK, emails: [email protected]; [email protected]

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.

Type
Papers
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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