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

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

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: [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]
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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.

References

Addison, J., Howison, S. D. & King, J. (2006) Ray methods for free boundary problems. Q. Appl. Math. 64(1), 41–59.CrossRefGoogle Scholar
Antonelli, F., Kohatsu-Higa, A. (2002) Rate of convergence of a particle method to the solution of the McKean–Vlasov equation. Ann. Appl. Probab. 12(2), 423–476.CrossRefGoogle Scholar
Borkar, V. & Suresh Kumar, K. (2010) McKean–Vlasov limit in portfolio optimization. Stochastic Anal. Appl. 28(5), 884–906.CrossRefGoogle Scholar
Bossy, M. & Talay, D. (1997) A stochastic particle method for the McKean-Vlasov and the Burgers equation. Math. Comput. 66(217), 157–192.CrossRefGoogle Scholar
Brunner, H. (1985) The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes. Math. Comput. 45(172), 417–437.CrossRefGoogle Scholar
Bujok, K. & Reisinger, C. (2012) Numerical valuation of basket credit derivatives in structural jump-diffusion models. J. Comput. Finance 15(4), 115.CrossRefGoogle Scholar
Bush, N., Hambly, B. M., Haworth, H., Jin, L. & Reisinger, C. (2011) Stochastic evolution equations in portfolio credit modelling. SIAM J. Financial Math. 2(1), 627–664.CrossRefGoogle Scholar
Cáceres, M. J., Carrillo, J. A. & Perthame, B. (2011) Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states. J. Math. Neurosci. 1(1), 7.CrossRefGoogle ScholarPubMed
Carrillo, J. A., González, M. d. M., Gualdani, M. P. & Schonbek, M. E. (2013) Classical solutions for a nonlinear Fokker–Planck equation arising in computational neuroscience. Commun. Partial Differ. Equ. 38(3), 385–409.CrossRefGoogle Scholar
David, A. & Lehar, A. (2014) Why are banks highly interconnected? Available at SSRN 1108870.Google Scholar
Delarue, F., Inglis, J., Rubenthaler, S. & Tanré, E. (2015a) Global solvability of a networked integrate-and-fire model of McKean–Vlasov type. Ann. Appl. Probab. 25(4), 2096–2133.CrossRefGoogle Scholar
Delarue, F., Inglis, J., Rubenthaler, S. & Tanré, E. (2015b) Particle systems with a singular mean-field self-excitation. Application to neuronal networks. Stochastic Processes Appl. 125(6), 2451–2492.CrossRefGoogle Scholar
Delarue, F., Nadtochiy, S. & Shkolnikov, M. (2019) Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness. arXiv preprint arXiv:1902.05174.Google Scholar
Dewynne, J. N., Howison, S. D., Ockendon, J. R. & Xie, W. (1989) Asymptotic behavior of solutions to the Stefan problem with a kinetic condition at the free boundary. ANZIAM J. 31(1), 81–96.Google Scholar
Eisenberg, L. & Noe, T. H. (2001) Systemic risk in financial systems. Manage. Sci. 47(2), 236–249.CrossRefGoogle Scholar
Fasano, A., Primicerio, M. & Hadeler, K. P. (1983) A critical case for the solvability of Stefan-like problems. Math. Methods Appl. Sci. 5(1), 84–96.CrossRefGoogle Scholar
Fasano, A., Primicerio, M., Howison, S. D. & Ockendon, J. R. (1989) On the singularities of one-dimensional Stefan problems with supercooling. In: Mathematical Models for Phase Change Problems, International Series of Numerical Mathematics Vol. 88, Birkhäuser, Basel, pp. 215–226.CrossRefGoogle Scholar
Fasano, A., Primicerio, M., Howison, S. D. & Ockendon, J. R. (1990) Some remarks on the regularization of supercooled one-phase Stefan problems in one dimension. Q. Appl. Math. 48(1), 153–168.CrossRefGoogle Scholar
Gobet, E. & Pagliarani, S. (2018) Analytical approximations of non-linear SDEs of McKean–Vlasov type. J. Math. Anal. Appl. 466(1), 71–106.CrossRefGoogle Scholar
Götz, I. G., Primicerio, M. & Velázquez, J. J. L. (2002) Asymptotic behaviour (t→+0) of the interface for the critical case of undercooled Stefan problem. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 13(2), 143–148.Google Scholar
Hairer, E., Lubich, C. & Schlichte, M. (1985) Fast numerical solution of nonlinear Volterra convolution equations. SIAM J. Sci. Stat. Comput. 6(3), 532–541.CrossRefGoogle Scholar
Haji-Ali, A.-L. & Tempone, R. (2018) Multilevel and multi-index Monte Carlo methods for the McKean–Vlasov equation. Stat. Comput. 28(4), 923–935.CrossRefGoogle Scholar
Hambly, B., Ledger, S. & Sojmark, A. (2019) A McKean–Vlasov equation with positive feedback and blow-ups. Ann. Appl. Probab. 29(4), 2338–2373.CrossRefGoogle Scholar
Hambly, B. & Sojmark, A. (2019) An SPDE model for systemic risk with endogenous contagion. Finance Stoch. 23(3), 535–594.CrossRefGoogle Scholar
Haworth, H. & Reisinger, C. (2007) Modeling basket credit default swaps with default contagion. J. Credit Risk 3(4), 31–67.CrossRefGoogle Scholar
Haworth, H., Reisinger, C. & Shaw, W. (2008) Modelling bonds and credit default swaps using a structural model with contagion. Quant. Finance 8(7), 669–680.CrossRefGoogle Scholar
Herrero, M. A. & Velázquez, J. J. L. (1996) Singularity formation in the one-dimensional supercooled Stefan problem. Eur. J. Appl. Math. 7(2), 119–150.CrossRefGoogle Scholar
Howison, S. D., Ockendon, J. R. & Lacey, A. A. (1985) Singularity development in moving-boundary problems. Q. J. Mech. Appl. Math. 38(3), 343–360.CrossRefGoogle Scholar
Howison, S. D. & Xie, W. (1989) Kinetic undercooling regularisation of supercooled Stefan problems. In: Mathematical Models for Phase Change Problems, International Series of Numerical Mathematics, Vol. 88, Birkhäuser, Basel, pp. 215–226.Google Scholar
Huang, M., Malhamé, R. P., Caines, P. E., et al. (2006) Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–252.Google Scholar
Ichiba, T., Ludkovski, M. & Sarantsev, A. (2019). Dynamic contagion in a banking system with births and defaults. Ann Finance 15(4), 489–538.CrossRefGoogle Scholar
Itkin, A. & Lipton, A. (2015) Efficient solution of structural default models with correlated jumps and mutual obligations. Int. J. Comput. Math. 92(12), 2380–2405.CrossRefGoogle Scholar
Itkin, A. & Lipton, A. (2017) Structural default model with mutual obligations. Rev. Derivatives Res. 20, 15–46.CrossRefGoogle Scholar
Kac, M. (1956) Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. 3, University of California Press Berkeley and Los Angeles, California, pp. 171–197.Google Scholar
Kaushansky, V., Lipton, A. & Reisinger, C. (2018a) Numerical analysis of an extended structural default model with mutual liabilities and jump risk. J. Comput. Sci. 24, 218–231.CrossRefGoogle Scholar
Kaushansky, V., Lipton, A. & Reisinger, C. (2018b) Transition probability of Brownian motion in the octant and its application to default modelling. Appl. Math. Finance. 25(5–6), 435–465.CrossRefGoogle Scholar
Kaushansky, V. & Reisinger, C. (2019) Simulation of particle systems interacting through hitting times. Discrete Continuous Dyn. Syst. Ser. B, 24(10): 5481–5502.Google Scholar
King, J. R. & Evans, J. D. (2005) Regularization by kinetic undercooling of blow-up in the ill-posed Stefan problem. SIAM J. Appl. Math. 65(5), 1677–1707.CrossRefGoogle Scholar
Kolk, M. & Pedas, A. (2009) Numerical solution of Volterra integral equations with weakly singular kernels which may have a boundary singularity. Math. Modell. Anal. 14(1), 79–89.CrossRefGoogle Scholar
Kolk, M. & Pedas, A. (2013) Numerical solution of Volterra integral equations with singularities. Front. Math. China 8(2), 239–259.CrossRefGoogle Scholar
Kolk, M., Pedas, A. & Vainikko, G. (2009) High-order methods for Volterra integral equations with general weak singularities. Numer. Funct. Anal. Optim. 30(9–10), 1002–1024.CrossRefGoogle Scholar
Ledger, S. & Sojmark, A. (2018) At the mercy of the common noise: blow-ups in a conditional McKean–Vlasov problem. arXiv preprint arXiv:1807.05126.Google Scholar
Linz, P. (1985) Analytical and Numerical Methods for Volterra Equations. SIAM Philadelphia.CrossRefGoogle Scholar
Lipton, A. (2001) Mathematical Methods For Foreign Exchange: A Financial Engineer’s Approach. World Scientific Publishing, Singapore.CrossRefGoogle Scholar
Lipton, A. (2016) Modern monetary circuit theory, stability of interconnected banking network, and balance sheet optimization for individual banks. Int. J. Theor. Appl. Finance 19(6). doi: 10.1142/S0219024916500345.CrossRefGoogle Scholar
Lipton, A. & Sepp, A. (2009) Credit value adjustment for credit default swaps via the structural default model. J. Credit Risk 5(2), 123–146.CrossRefGoogle Scholar
Maleknejad, K., Hashemizadeh, E. & Ezzati, R. (2011) A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation. Commun. Nonlinear Sci. Numer. Simul. 16(2), 647–655.CrossRefGoogle Scholar
Maleknejad, K., Sohrabi, S. & Rostami, Y. (2007) Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials. Appl. Math. Comput. 188(1), 123–128.Google Scholar
McKean, H. P. (1966) A class of Markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. 56(6), 1907–1911.CrossRefGoogle ScholarPubMed
Merton, R. C. (1974) On the pricing of corporate debt: the risk structure of interest rates. J. Finance 29(2), 449–470.Google Scholar
Nadtochiy, S. & Shkolnikov, M. (2019) Particle systems with singular interaction through hitting times: application in systemic risk modeling. 29(1), 89–129.CrossRefGoogle Scholar
Nadtochiy, S. & Shkolnikov, M. (2018) Mean field systems on networks, with singular interaction through hitting times. arXiv preprint arXiv:1807.02015.Google Scholar
Noble, B. (1969) Instability when solving Volterra integral equations of the second kind by multistep methods. In: Conference on the Numerical Solution of Differential Equations, Springer, pp. 23–39.CrossRefGoogle Scholar
Oleinik, O. A., Primicerio, M. & Radkevich, E. V. (1993) Stefan-like problems. Meccanica 28(2), 129–143.CrossRefGoogle Scholar
Peskir, G. (2002) On integral equations arising in the first-passage problem for Brownian motion. J. Integral Equ. Appl. 14(4), 397–423.CrossRefGoogle Scholar
Reis, G. d., Smith, G. & Tankov, P. (2018) Importance sampling for McKean-Vlasov SDEs. arXiv preprint arXiv:1803.09320.Google Scholar
Ricketson, L. (2015) A multilevel Monte Carlo method for a class of McKean–Vlasov processes. arXiv preprint arXiv:1508.02299.Google Scholar
Rubinstein, L. I. (1971) The Stefan Problem, Translations of Mathematical Monographs, Vol. 27, American Mathematical Society, Providence, RI.Google Scholar
Szpruch, L., Tan, S. & Tse, A. (2017) Iterative particle approximation for McKean–Vlasov SDEs with application to Multilevel Monte Carlo estimation. arXiv preprint arXiv:1706.00907.Google Scholar
Tikhonov, A. N. & Samarskii, A. A. (1963) Equations of Mathematical Physics. Dover Publications, New York. English Translation.Google Scholar
Watson, N. A. (2012) Introduction to Heat Potential Theory, Mathematical Surveys and Monographs, Vol. 182. Providence, Rhode Island.Google Scholar