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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

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

Received  June 2018 Published  October 2019 Early access  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, 2019, 24 (10) : 5481-5502. doi: 10.3934/dcdsb.2019067
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##### References:
(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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