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A statistical approach for simulating the density solution of a McKean-Vlasov equation

Marc Hoffmann, Yating Liu

Abstract

We prove optimal convergence results of a stochastic particle method for computing the classical solution of a multivariate McKean-Vlasov equation, when the measure variable is in the drift, following the classical approach of [BT97, AKH02]. Our method builds upon adaptive nonparametric results in statistics that enable us to obtain a data-driven selection of the smoothing parameter in a kernel type estimator. In particular, we generalise the Bernstein inequality of [DMH21] for mean-field McKean-Vlasov models to interacting particles Euler schemes and obtain sharp deviation inequalities for the estimated classical solution. We complete our theoretical results with a systematic numerical study, and gather empirical evidence of the benefit of using high-order kernels and data-driven smoothing parameters.

A statistical approach for simulating the density solution of a McKean-Vlasov equation

Abstract

We prove optimal convergence results of a stochastic particle method for computing the classical solution of a multivariate McKean-Vlasov equation, when the measure variable is in the drift, following the classical approach of [BT97, AKH02]. Our method builds upon adaptive nonparametric results in statistics that enable us to obtain a data-driven selection of the smoothing parameter in a kernel type estimator. In particular, we generalise the Bernstein inequality of [DMH21] for mean-field McKean-Vlasov models to interacting particles Euler schemes and obtain sharp deviation inequalities for the estimated classical solution. We complete our theoretical results with a systematic numerical study, and gather empirical evidence of the benefit of using high-order kernels and data-driven smoothing parameters.
Paper Structure (17 sections, 9 theorems, 126 equations, 11 figures, 1 table)

This paper contains 17 sections, 9 theorems, 126 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results and organisation of the paper
  4. Main results
  5. Notation and assumptions
  6. Construction of an estimator of $\mu_T(x)$
  7. Convergence results
  8. Nonlinearity of the drift in the measure argument
  9. Numerical implementation
  10. The case of a linear interaction
  11. A double layer potential
  12. The Burgers equation in dimension $d=1$
  13. Proofs
  14. Proof of Theorem \ref{['thm: deviation']}
  15. Proof of Theorem \ref{['thm: error 1']}
  16. ...and 2 more sections

Key Result

Theorem 2.6

Work under Assumptions ass: AI, ass: AII and ass: AIII 1. Let $x \in \mathbb{R}^d$, $N\geq 2$, $\varepsilon >0$ and $\mathfrak C>0$. For any $0 < h < \min(\,h^\star(\tfrac{1}{3}|K|_{L^1}^{-1}\varepsilon\,), \,\mathfrak CN^{-1})$ and $0 < \eta < \eta^\star(\tfrac{1}{3}\varepsilon, \mu_T)$, we have: where $\kappa_1$ is a constant depending on $\mathfrak{C}$ and $\mathfrak{M}$, and $\kappa_2$ depen

Figures (11)

  • Figure 1: Monte-Carlo (for 30 repeated samples) strong error for different kernel orders: $\log_2 \mathcal{E}_N$ as a function of $\log_2 N$ for $\ell = 1$ (purple), $\ell = 3$ (blue), $\ell = 5$ (red), $\ell = 7$ (orange), $\ell = 9$ (green). We see that a polynomial error in $N$ is compatible with the data.
  • Figure 2: Least-square estimates of the slope $\alpha_\ell$ of $\log_2 \mathcal{E}_N = \alpha_\ell \log_2 N+\text{noise}$ in a linear model representation. We plot $\alpha_\ell$ as a function of the order $\ell$ of the kernel. We see that a higher order $\ell$ for the choice of the kernel systematically improves on the error rate, as predicted by the statistical bias-variance analysis.
  • Figure 3: Plot of the potential $U$ (red) and its derivative $U'$ (blue).
  • Figure 4: The graph of $x \mapsto \widehat{\mu}_t^{N, h, \widehat{\eta}^N(t,x)}(x)$. The domain $x \in [-4,4]$ is computed over a discrete grid of $2000$ points, i.e. mesh $4 \cdot 10^{-3}$) for $N = 2^5$ (Left) and $N = 2^{10}$ (Right).
  • Figure 5: Same experiment as in Figure \ref{['fig: doublelayer_bandwidth_effect']} mimicking an asymptotic behaviour of the procedure for $N=2^{15}$.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Theorem 2.9
  • Corollary 2.10
  • ...and 6 more