Figure 1.
(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 $
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October
2019, 24(10): 5481-5502.
doi: 10.3934/dcdsb.2019067
Simulation of a simple particle system interacting through hitting times
University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, UK, OX2 6GG |
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.
Keywords:
McKean-Vlasov equations,
particle method,
timestepping scheme,
Brownian bridge,
mean-field problems.
Mathematics Subject Classification:
Primary: 65C35; Secondary: 60K35.
Citation:
Vadim Kaushansky, Christoph Reisinger. Simulation of a simple particle system interacting through hitting times. Discrete and Continuous Dynamical Systems - B,
2019, 24
(10)
: 5481-5502.
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. |
show all references
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. |
Figure 8.
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
Figure 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.
Figure 4.
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
Figure 6.
(a) Loss process computed using Algorithm 1 for different $ n $ ; (b) error as a function of $ t $
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