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Simulation of a simple particle system interacting through hitting times

  • * Corresponding author: Vadim Kaushansky

    * Corresponding author: Vadim Kaushansky 

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

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  • 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.

    Mathematics Subject Classification: Primary: 65C35; Secondary: 60K35.


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  • 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 $

    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 3.  (a) $ L_t $ and (b) $ L'_t $ for different values of $ \alpha $

    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 5.  (a) $ L_t $ and (b) $ L'_t $ for different values of $ \alpha $

    Figure 6.  (a) Loss process computed using Algorithm 1 for different $ n $; (b) error as a function of $ t $

    Figure 7.  Convergence rate: at (a) $ T = 0.001 $, (b) $ T = 0.008 $

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