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Particle Method for the McKean-Vlasov equation with common noise

Théophile Le Gall

TL;DR

This work addresses numerical simulation of the McKean-Vlasov equation with common noise by integrating an Euler time discretization with a particle method that estimates the conditional law via the empirical distribution. Under Hölder time regularity and Lipschitz continuity in state and measure, the authors establish convergence rates for both the Euler scheme and the particle method, extending known results for the standard MKV equation to the common-noise setting. The theoretical findings are complemented by numerical simulations on a modified conditional Ornstein-Uhlenbeck process and an interbank market model, illustrating the practical accuracy and providing empirical convergence rates. Overall, the paper provides a rigorous numerical framework for mean-field systems with shared randomness, enabling reliable simulations in finance and related fields.

Abstract

This paper studies the numerical simulation of the solution to the McKean-Vlasov equation with common noise. We begin by discretizing the solution in time using the Euler scheme, followed by spatial discretization through the particle method, inspired by the propagation of chaos property. Assuming H{ö}lder continuity in time, as well as Lipschitz continuity in the state and measure arguments of the coefficient functions, we establish the convergence rate of the Euler scheme and the particle method. These results extend those for the standard McKean-Vlasov equation without common noise. Finally, we present two simulation examples : a modified conditional Ornstein Uhlenbeck process with common noise and an interbank market model.

Particle Method for the McKean-Vlasov equation with common noise

TL;DR

This work addresses numerical simulation of the McKean-Vlasov equation with common noise by integrating an Euler time discretization with a particle method that estimates the conditional law via the empirical distribution. Under Hölder time regularity and Lipschitz continuity in state and measure, the authors establish convergence rates for both the Euler scheme and the particle method, extending known results for the standard MKV equation to the common-noise setting. The theoretical findings are complemented by numerical simulations on a modified conditional Ornstein-Uhlenbeck process and an interbank market model, illustrating the practical accuracy and providing empirical convergence rates. Overall, the paper provides a rigorous numerical framework for mean-field systems with shared randomness, enabling reliable simulations in finance and related fields.

Abstract

This paper studies the numerical simulation of the solution to the McKean-Vlasov equation with common noise. We begin by discretizing the solution in time using the Euler scheme, followed by spatial discretization through the particle method, inspired by the propagation of chaos property. Assuming H{ö}lder continuity in time, as well as Lipschitz continuity in the state and measure arguments of the coefficient functions, we establish the convergence rate of the Euler scheme and the particle method. These results extend those for the standard McKean-Vlasov equation without common noise. Finally, we present two simulation examples : a modified conditional Ornstein Uhlenbeck process with common noise and an interbank market model.

Paper Structure

This paper contains 12 sections, 15 theorems, 84 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Probabilistic settings
  3. Construction of the Euler scheme and particle method
  4. Assumptions and main results
  5. Organization of the paper
  6. Preliminary results
  7. Convergence rate of the Euler scheme
  8. Convergence rate of the particle system
  9. Numerical simulations
  10. Conditional Ornstein-Uhlenbeck process with common noise
  11. Interbank market model
  12. Appendix

Key Result

Proposition 1.1

Given a random variable $X: (\Omega, \mathcal{F}, \mathbb{P})\rightarrow (\mathop{\mathrm{\mathbb{R}}}\nolimits^d, \mathcal{B}(\mathop{\mathrm{\mathbb{R}}}\nolimits^d))$, the mapping $\mathcal{L}^1(X)$ defined by eq:def-L1X is almost surely well defined under $\mathbb{P}^0$, and forms a random varia

Figures (4)

  • Figure 1: Log-log error \ref{['eq:approxi-l2-error']} between $(X_t)_{t\in[0,T]}$ and $(\bar{X}_{t_m})_{0\leq m\leq M}$ (slope = -0.35)
  • Figure 2: Log-log error \ref{['eq:approxi-density-error']} between $\hat{\mu}^{N,h,\eta}_T$ and $\mu_T$ (slope = -0.4)
  • Figure 3: Log-log error of $\mathcal{E}_N$ defined by \ref{['eq:interbank-error']} (slope = -0.4)
  • Figure 4: 10 paths of particles $(X^{i,N}_t)_{t\in[0,T]}$

Theorems & Definitions (27)

  • Proposition 1.1: Carmona_Delarue2
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1: Lemma 2.5 in Carmona_Delarue2
  • Remark 2.2: Remark 2.3 in Carmona_Delarue2
  • Lemma 2.3: General Minkowski inequality
  • Lemma 2.4: Burkölder-Davis-Gundy inequality
  • Lemma 2.5: 'A la Gronwall' Lemma
  • Lemma 2.6
  • ...and 17 more