A novel numerical method for mean field stochastic differential equation

TL;DR

The authors address numerical approximation of mean field SDEs by transforming the problem into a nonlinear Fokker-Planck equation for the solution density $p(t,x)$. They propose truncating the domain to a bounded region, solving a stable explicit-implicit finite difference scheme to obtain a numerical density $p_{\mathcal{D}}$, and then solving an SDE driven by this density to approximate the mean field dynamics. They provide error estimates linking PDE discretization, domain truncation, and SDE approximation errors, and validate the approach with numerical experiments that show favorable convergence and accuracy compared to particle-based methods. This density-based framework reduces the computational burden associated with simulating high-dimensional interacting particle systems while achieving high-precision approximations of the mean field behavior. The method is particularly advantageous when the law plays a central role, enabling efficient computation of trajectories and moments without large-scale Monte Carlo sampling.

Abstract

In this paper, we propose a novel method to approximate the mean field stochastic differential equation by means of approximating the density function via Fokker-Planck equation. We construct a well-posed truncated Fokker-Planck equation whose solution is an approximation to the density function of solution to the mean field stochastic differential equation. We also apply finite difference method to approximate the truncated Fokker-Planck equation and derive error estimates. We use the numerical density function to replace the true measure in mean field stochastic differential equation and set up a stochastic differential equation to approximate the mean field one. Meanwhile, we derive the corresponding error estimates. Finally, we present several numerical experiments to illustrate the theoretical analysis.

Figures (9)

• Figure 1: $\log-\log$ error.
• Figure 3: Error and convergence order.
• Figure 4: Expectation and variance.
• Figure 4: $\log-\log$ error.
• Figure 7: Error and convergence order.

Theorems & Definitions (10)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof
• Example 1
• Example 2
• Example 3