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October 2017 A stochastic McKean–Vlasov equation for absorbing diffusions on the half-line
Ben Hambly, Sean Ledger
Ann. Appl. Probab. 27(5): 2698-2752 (October 2017). DOI: 10.1214/16-AAP1256

Abstract

We study a finite system of diffusions on the half-line, absorbed when they hit zero, with a correlation effect that is controlled by the proportion of the processes that have been absorbed. As the number of processes in the system becomes large, the empirical measure of the population converges to the solution of a nonlinear stochastic heat equation with Dirichlet boundary condition. The diffusion coefficients are allowed to have finitely many discontinuities (piecewise Lipschitz) and we prove pathwise uniqueness of solutions to the limiting stochastic PDE. As a corollary, we obtain a representation of the limit as the unique solution to a stochastic McKean–Vlasov problem. Our techniques involve energy estimation in the dual of the first Sobolev space, which connects the regularity of solutions to their boundary behaviour, and tightness calculations in the Skorokhod M1 topology defined for distribution-valued processes, which exploits the monotonicity of the loss process $L$. The motivation for this model comes from the analysis of large portfolio credit problems in finance.

Citation

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Ben Hambly. Sean Ledger. "A stochastic McKean–Vlasov equation for absorbing diffusions on the half-line." Ann. Appl. Probab. 27 (5) 2698 - 2752, October 2017. https://doi.org/10.1214/16-AAP1256

Information

Received: 1 April 2016; Revised: 1 October 2016; Published: October 2017
First available in Project Euclid: 3 November 2017

zbMATH: 1379.60068
MathSciNet: MR3719945
Digital Object Identifier: 10.1214/16-AAP1256

Subjects:
Primary: 60F15, 60H15, 60H30

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 5 • October 2017
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