# American Institute of Mathematical Sciences

## Simulation of a simple particle system interacting through hitting times

 University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, UK, OX2 6GG

The frst author gratefully acknowledges support from the Economic and Social Research Council and Bank of America Merrill Lynch.

Received  June 2018 Published  April 2019

We develop an Euler-type particle method for the simulation of a McKean–Vlasov equation arising from a mean-field model with positive feedback from hitting a boundary. Under assumptions on the parameters which ensure differentiable solutions, we establish convergence of order $1/2$ in the time step. Moreover, we give a modification of the scheme using Brownian bridges and local mesh refinement, which improves the order to $1$. We confirm our theoretical results with numerical tests and empirically investigate cases with blow-up.

Citation: Vadim Kaushansky, Christoph Reisinger. Simulation of a simple particle system interacting through hitting times. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019067
##### References:
 [1] F. Antonelli and A. Kohatsu-Higa, Rate of convergence of a particle method to the solution of the McKean–Vlasov equation, The Annals of Applied Probability, 12 (2002), 423-476. doi: 10.1214/aoap/1026915611. [2] S. Asmussen, P. Glynn and J. Pitman, Discretization error in simulation of one-dimensional reflecting Brownian motion, The Annals of Applied Probability, 5 (1995), 875-896. doi: 10.1214/aoap/1177004597. [3] K. Borovkov and A. Novikov, Explicit bounds for approximation rates of boundary crossing probabilities for the Wiener process, Journal of Applied Probability, 42 (2005), 82-92. doi: 10.1239/jap/1110381372. [4] M. Bossy and D. Talay, A stochastic particle method for the McKean-Vlasov and the Burgers equation, Mathematics of Computation, 66 (1997), 157-192. doi: 10.1090/S0025-5718-97-00776-X. [5] M. J. Cáceres, J. A. Carrillo and B. Perthame, Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states, The Journal of Mathematical Neuroscience, 1 (2011), Art. 7, 33 pp. doi: 10.1186/2190-8567-1-7. [6] J. A. Carrillo, M. d. M. González, M. P. Gualdani and M. E. Schonbek, Classical solutions for a nonlinear Fokker–Planck equation arising in computational neuroscience, Communications in Partial Differential Equations, 38 (2013), 385-409. doi: 10.1080/03605302.2012.747536. [7] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, Global solvability of a networked integrate-and-fire model of McKean–Vlasov type, The Annals of Applied Probability, 25 (2015), 2096-2133. doi: 10.1214/14-AAP1044. [8] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, Particle systems with a singular mean-field self-excitation. Application to neuronal networks, Stochastic Processes and their Applications, 125 (2015), 2451-2492. doi: 10.1016/j.spa.2015.01.007. [9] G. Dos Reis, G. Smith and P. Tankov, Importance sampling for McKean-Vlasov SDEs, 2018, arXiv: 1803.09320. [10] T. A. Driscoll, N. Hale and L. N. Trefethen, Chebfun guide, Pafnuty Publ. [11] P. Glasserman, Monte Carlo Methods in Financial Engineering, vol. 53, Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2004. [12] B. Hambly, S. Ledger and A. Sojmark, A McKean–Vlasov equation with positive feedback and blow-ups, arXiv: 1801.07703. [13] A. Lipton, Modern monetary circuit theory, stability of interconnected banking network, and balance sheet optimization for individual banks, International Journal of Theoretical and Applied Finance, 19 (2016), 1650034, 57 pp. doi: 10.1142/S0219024916500345. [14] S. Nadtochiy and M. Shkolnikov, Particle systems with singular interaction through hitting times: Application in systemic risk modeling, The Annals of Applied Probability, 29 (2019), 89-129. doi: 10.1214/18-AAP1403. [15] L. Ricketson, A multilevel Monte Carlo method for a class of McKean–Vlasov processes, arXiv: 1508.02299. [16] L. Szpruch, S. Tan and A. Tse, Iterative particle approximation for McKean–Vlasov SDEs with application to Multilevel Monte Carlo estimation, arXiv: 1706.00907.

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##### References:
 [1] F. Antonelli and A. Kohatsu-Higa, Rate of convergence of a particle method to the solution of the McKean–Vlasov equation, The Annals of Applied Probability, 12 (2002), 423-476. doi: 10.1214/aoap/1026915611. [2] S. Asmussen, P. Glynn and J. Pitman, Discretization error in simulation of one-dimensional reflecting Brownian motion, The Annals of Applied Probability, 5 (1995), 875-896. doi: 10.1214/aoap/1177004597. [3] K. Borovkov and A. Novikov, Explicit bounds for approximation rates of boundary crossing probabilities for the Wiener process, Journal of Applied Probability, 42 (2005), 82-92. doi: 10.1239/jap/1110381372. [4] M. Bossy and D. Talay, A stochastic particle method for the McKean-Vlasov and the Burgers equation, Mathematics of Computation, 66 (1997), 157-192. doi: 10.1090/S0025-5718-97-00776-X. [5] M. J. Cáceres, J. A. Carrillo and B. Perthame, Analysis of nonlinear noisy integrate & fire neuron models: Blow-up and steady states, The Journal of Mathematical Neuroscience, 1 (2011), Art. 7, 33 pp. doi: 10.1186/2190-8567-1-7. [6] J. A. Carrillo, M. d. M. González, M. P. Gualdani and M. E. Schonbek, Classical solutions for a nonlinear Fokker–Planck equation arising in computational neuroscience, Communications in Partial Differential Equations, 38 (2013), 385-409. doi: 10.1080/03605302.2012.747536. [7] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, Global solvability of a networked integrate-and-fire model of McKean–Vlasov type, The Annals of Applied Probability, 25 (2015), 2096-2133. doi: 10.1214/14-AAP1044. [8] F. Delarue, J. Inglis, S. Rubenthaler and E. Tanré, Particle systems with a singular mean-field self-excitation. Application to neuronal networks, Stochastic Processes and their Applications, 125 (2015), 2451-2492. doi: 10.1016/j.spa.2015.01.007. [9] G. Dos Reis, G. Smith and P. Tankov, Importance sampling for McKean-Vlasov SDEs, 2018, arXiv: 1803.09320. [10] T. A. Driscoll, N. Hale and L. N. Trefethen, Chebfun guide, Pafnuty Publ. [11] P. Glasserman, Monte Carlo Methods in Financial Engineering, vol. 53, Stochastic Modelling and Applied Probability, Springer-Verlag, New York, 2004. [12] B. Hambly, S. Ledger and A. Sojmark, A McKean–Vlasov equation with positive feedback and blow-ups, arXiv: 1801.07703. [13] A. Lipton, Modern monetary circuit theory, stability of interconnected banking network, and balance sheet optimization for individual banks, International Journal of Theoretical and Applied Finance, 19 (2016), 1650034, 57 pp. doi: 10.1142/S0219024916500345. [14] S. Nadtochiy and M. Shkolnikov, Particle systems with singular interaction through hitting times: Application in systemic risk modeling, The Annals of Applied Probability, 29 (2019), 89-129. doi: 10.1214/18-AAP1403. [15] L. Ricketson, A multilevel Monte Carlo method for a class of McKean–Vlasov processes, arXiv: 1508.02299. [16] L. Szpruch, S. Tan and A. Tse, Iterative particle approximation for McKean–Vlasov SDEs with application to Multilevel Monte Carlo estimation, arXiv: 1706.00907.
(a) $L_t$ for different $\alpha$ near the jump; (b) distribution of $Y_T$ for $Y_T > 0$ before and after the jump. Fitted by kernel density estimation with normal kernel for $N = 10^7$
Convergence rate for $d_1(L_t, \tilde{L}_t)$, $d_2(L_t, \tilde{L}_t)$, $d_3(L_t, \tilde{L}_t)$ for (a) Algorithm 1, (b) Algorithm 2
Error of the loss process at $t=T$ for $\frac{1}{Y_0} \sim \exp(1)$ : %depending on time, (a) for increasing number $n$ of timesteps; (b) for increasing number $N$ of samples, %with $\frac{1}{Y_0} \sim \exp(1)$ both for Algorithms 1 and 2.
(a) $L_t$ and (b) $L'_t$ for different values of $\alpha$
Error of the loss process at $t = T$ for ${{Y}_{0}}\tilde{\ }\text{Gammadistr}(1+\beta ,1/2)$: (a) for increasing number $n$ of timesteps; (b) for increasing number $N$ of samples, both for Algorithms 1 and 2
(a) $L_t$ and (b) $L'_t$ for different values of $\alpha$
(a) Loss process computed using Algorithm 1 for different $n$; (b) error as a function of $t$
Convergence rate: at (a) $T = 0.001$, (b) $T = 0.008$
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