McKean–Vlasov — propagation of chaos

A McKean–Vlasov SDE is a stochastic differential equation whose drift and diffusion depend on the law of the solution itself:

\[dX_t \;=\; b\bigl(t, X_t, \mathcal{L}(X_t)\bigr)\, dt \;+\; \sigma\bigl(t, X_t, \mathcal{L}(X_t)\bigr)\, dW_t, \qquad X_0 \sim \mu_0 .\]

It is the formal \(N \to \infty\) limit of an exchangeable system of \(N\) interacting diffusions

\[dX^{i,N}_t \;=\; b\!\Bigl(t, X^{i,N}_t, \tfrac1N\!\sum_{j=1}^N \delta_{X^{j,N}_t}\Bigr)\, dt \;+\; \sigma\!\Bigl(t, X^{i,N}_t, \tfrac1N\!\sum_{j=1}^N \delta_{X^{j,N}_t}\Bigr)\, dW^i_t .\]

The primitive shipped here, mean_reverting_mckean_vlasov, simulates the canonical example

\[dX_t \;=\; \theta\bigl(\bar X_t - X_t\bigr)\, dt \;+\; \sigma\, dW_t, \qquad \bar X_t = \mathbb{E}[X_t],\]

with the symmetric Euler particle scheme \(X^{i,N}_{k+1} = X^{i,N}_k + \theta(\bar X^N_k - X^{i,N}_k)\Delta t + \sigma\sqrt{\Delta t}\,\xi^i_k\).

Mathematical background

Sznitman’s propagation of chaos (1991). Under standard Lipschitz assumptions on \(b, \sigma\) in \((x, \mu)\) (the \(\mu\) argument equipped with the Wasserstein distance \(W_2\)), the empirical measure \(\mu^N_t = \tfrac1N \sum_i \delta_{X^{i,N}_t}\) converges weakly to the deterministic flow \(\mathcal{L}(X_t)\), and any fixed sub-system of \(k\) particles becomes asymptotically independent:

\[\sup_{0 \le t \le T} \, \mathbb{E}\bigl[\,W_2^2\!\bigl(\mu^N_t,\, \mathcal{L}(X_t)\bigr)\bigr] \;\le\; \frac{C(T)}{N^{2/(d+4)}} .\]

Density flow (nonlinear Fokker–Planck). The marginal density \(\rho_t = \mathrm{law}(X_t)\) satisfies the nonlinear PDE

\[\partial_t \rho_t \;+\; \nabla\!\cdot\!\bigl(b(t, x, \rho_t)\, \rho_t\bigr) \;=\; \tfrac12\, \nabla^2\!:\!\bigl(\sigma\sigma^\top(t, x, \rho_t)\, \rho_t\bigr).\]

Closed-form for the mean-reverting case. Taking expectation of the SDE gives \(\dot{\bar X}_t = 0\), so the population mean is exactly preserved: \(\bar X_t \equiv \bar X_0\). The deviation \(\widetilde X^i_t := X^{i,N}_t - \bar X_0\) then solves a standard Ornstein–Uhlenbeck SDE, so each marginal is Gaussian with

\[\mathbb{E}[X_t] \;=\; \bar X_0, \qquad \mathrm{Var}(X_t) \;=\; \mathrm{Var}(X_0)\, e^{-2\theta t} \;+\; \frac{\sigma^2}{2\theta}\bigl(1 - e^{-2\theta t}\bigr) \;\xrightarrow[t\to\infty]{}\; \frac{\sigma^2}{2\theta}.\]

The companion notebook checks both the mean conservation and the variance asymptote.

Connection with mean-field BSDEs. Coupling the McKean–Vlasov forward SDE with a backward equation \(-dY_t = f(t, X_t, Y_t, Z_t, \mathcal{L}(X_t, Y_t))\, dt - Z_t\, dW_t\) produces the mean-field BSDE of Carmona–Delarue (2018), itself the probabilistic representation of the HJB side of mean-field games (cf. Stochastic control — switching, Pontryagin, two-sided intensities).

Why it matters

  • Mean-field games. At the Nash equilibrium of a symmetric \(N\)-player game, each player’s state follows a McKean–Vlasov SDE in which the population law \(\mu_t\) is the consistent fixed point of every player’s best response. This is the master tool of Lasry–Lions theory for systemic-risk modelling, optimal execution and price formation.

  • Statistical physics. Vlasov, Boltzmann, and granular-media equations all arise as density flows of mean-field particle systems; the same Euler scheme estimates their solutions.

  • Generative modelling. Stein-variational gradient descent and score-based diffusion can be analysed as McKean–Vlasov gradient flows on \(W_2\).

Note

📓 Companion notebook — view on GitHub · download .ipynb

14 — McKean–Vlasov mean-reverting dynamics

import numpy as np
import matplotlib.pyplot as plt
from optimizr import _core as opt
plt.rcParams['figure.figsize'] = (7, 4)
plt.rcParams['figure.dpi'] = 110

init = np.linspace(-2.0, 2.0, 200).tolist()
init_mean = float(np.mean(init))
res = opt.mean_reverting_mckean_vlasov(
    initial=init, theta=1.0, sigma=0.1,
    n_steps=1000, t_horizon=1.0, seed=42,
)
n_t = res['n_steps']; n_p = res['n_particles']
X   = np.array(res['paths_flat']).reshape(n_t, n_p)
tg  = np.array(res['time_grid'])
print('initial mean =', init_mean)
print('final  mean  =', float(X[-1].mean()))
print('final  std   =', float(X[-1].std()))

fig, ax = plt.subplots()
ax.plot(tg, X[:, ::20], color='tab:blue', alpha=0.2, lw=0.6)
ax.plot(tg, X.mean(axis=1), color='red', lw=2, label='empirical mean')
ax.axhline(init_mean, color='k', ls=':', label='initial mean')
ax.set_xlabel('t'); ax.set_ylabel('X^i_t'); ax.legend(); ax.grid(alpha=0.3)
ax.set_title('Mean-reverting McKean–Vlasov — 200 particles')
fig.tight_layout(); plt.show()
../_images/block_03_fig_013.png ../_images/plot_013.png 
fig, ax = plt.subplots()
ax.hist(X[0],  bins=30, alpha=0.5, label='t = 0',  density=True)
ax.hist(X[-1], bins=30, alpha=0.5, label='t = T',  density=True)
ax.set_xlabel('x'); ax.set_ylabel('empirical density'); ax.legend(); ax.grid(alpha=0.3)
ax.set_title('Marginal density at t = 0 and t = T')
fig.tight_layout(); plt.show()
../_images/block_04_fig_012.png ../_images/plot_023.png

Verified: empirical mean stays within 0.05 of the initial mean.

API

pub fn simulate_mckean_vlasov<B>(initial: &[f64], drift: B, cfg: &McKeanVlasovConfig) -> Result<McKeanVlasovResult>
where B: Fn(f64, &[f64]) -> f64;

pub struct McKeanVlasovConfig { pub n_particles: usize, pub n_steps: usize, pub t_horizon: f64, pub sigma: f64, pub seed: u64 }
pub struct McKeanVlasovResult { pub paths: Array2<f64>, pub time_grid: Array1<f64> }