Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tamed Euler approximation for fully superlinear growth McKean-Vlasov SDE and their particle systems: sharp rates for strong propagation of chaos, convergence and ergodicity

Simran Soni, Neelima, Chaman Kumar, Goncalo dos Reis

TL;DR

This work advances numerical analysis for McKean–Vlasov SDEs with fully superlinear growth by (i) proving well-posedness and moment bounds under Vlasov structure, (ii) establishing two Propagation of Chaos results with explicit rates, including a dimension-free $N^{-1/2}$ rate in the Vlasov-kernel setting, (iii) introducing a fully explicit taming Euler scheme that achieves strong convergence rate $1/2$ without requiring ellipticity or smoothness, and (iv) proving exponential ergodicity for the MV-SDE, the IPS, and the taming scheme. The dimension-free PoC result and the taming-based ergodic analysis are notable novelties, avoiding Kantorovich duality arguments in the PoC analysis and providing robust long-time stability for fully non-Lipschitz, measure-dependent dynamics. Together, these results enhance both theoretical understanding and practical computation for high-dimensional, mean-field systems with strong nonlinearity in both state and distribution arguments.

Abstract

We study McKean--Vlasov Stochastic Differential Equations (MV-SDEs) whose drift and diffusion coefficients are of superlinear growth in \textit{all} their variables thus also superlinear in the measure component (the meaning is specified in the body of the paper). We address the finite and infinite time horizon case. Our contribution is fourfold. (a) We establish well-posedness for this class of equations and the corresponding interacting particle system. (b) We prove two propagation of chaos results with explicit $L^2$-convergence rates: the first, is a general one where the rate degrades as the system's dimension $d$ increases; the second, attains the sharp rate $N^{-1/2}$ (in particle number $N$) uniformly over the dimension $d$ at the cost of a Vlasov kernel structure that is general and of superlinear growth for the measure dependency -- the latter's proof fully avoids the Kantorovich-Rubinstein duality argument. (c) Unlike existing works -- based on semi-implicit schemes or truncated Euler schemes -- we propose a fully explicit tamed Euler scheme that has reduced computational cost (comparatively). The explicit scheme is shown to converge in strong $L^p$-sense with rate $1/2$ (in timestep). (d) Lastly, we establish exponential ergodicity properties and long-time behavior for the MV-SDE, the corresponding interacting particle system, and the tamed scheme. The latter result is, to the best of our knowledge, fully novel.

Tamed Euler approximation for fully superlinear growth McKean-Vlasov SDE and their particle systems: sharp rates for strong propagation of chaos, convergence and ergodicity

TL;DR

This work advances numerical analysis for McKean–Vlasov SDEs with fully superlinear growth by (i) proving well-posedness and moment bounds under Vlasov structure, (ii) establishing two Propagation of Chaos results with explicit rates, including a dimension-free rate in the Vlasov-kernel setting, (iii) introducing a fully explicit taming Euler scheme that achieves strong convergence rate without requiring ellipticity or smoothness, and (iv) proving exponential ergodicity for the MV-SDE, the IPS, and the taming scheme. The dimension-free PoC result and the taming-based ergodic analysis are notable novelties, avoiding Kantorovich duality arguments in the PoC analysis and providing robust long-time stability for fully non-Lipschitz, measure-dependent dynamics. Together, these results enhance both theoretical understanding and practical computation for high-dimensional, mean-field systems with strong nonlinearity in both state and distribution arguments.

Abstract

We study McKean--Vlasov Stochastic Differential Equations (MV-SDEs) whose drift and diffusion coefficients are of superlinear growth in \textit{all} their variables thus also superlinear in the measure component (the meaning is specified in the body of the paper). We address the finite and infinite time horizon case. Our contribution is fourfold. (a) We establish well-posedness for this class of equations and the corresponding interacting particle system. (b) We prove two propagation of chaos results with explicit -convergence rates: the first, is a general one where the rate degrades as the system's dimension increases; the second, attains the sharp rate (in particle number ) uniformly over the dimension at the cost of a Vlasov kernel structure that is general and of superlinear growth for the measure dependency -- the latter's proof fully avoids the Kantorovich-Rubinstein duality argument. (c) Unlike existing works -- based on semi-implicit schemes or truncated Euler schemes -- we propose a fully explicit tamed Euler scheme that has reduced computational cost (comparatively). The explicit scheme is shown to converge in strong -sense with rate (in timestep). (d) Lastly, we establish exponential ergodicity properties and long-time behavior for the MV-SDE, the corresponding interacting particle system, and the tamed scheme. The latter result is, to the best of our knowledge, fully novel.
Paper Structure (16 sections, 21 theorems, 206 equations, 1 table)

This paper contains 16 sections, 21 theorems, 206 equations, 1 table.

Table of Contents

  1. Introduction
  2. Well-posedness of MV-SDEs: full superlinear growth under the Vlasov structure
  3. The particle systems and sharp rates for Propagation of Chaos
  4. Propagation of chaos -- The general Wasserstein-2 case
  5. Propagation of chaos -- the sharp (dimension-independent) rate result
  6. The tamed Euler scheme for the interacting particle system and its convergence rate
  7. Moment bounds
  8. Convergence rate of the scheme
  9. Ergodicity
  10. Ergodicity of the McKean--Vlasov SDE
  11. The Vlasov kernel case
  12. Ergodicity of the interacting particle system
  13. Ergodicity of the tamed Euler scheme
  14. Additional assumptions, remarks and preliminary results
  15. The main result
  16. ...and 1 more sections

Key Result

Theorem 2.1

Take $q> 0$ and $p_0~> 2(q+1)$. Let Assumptions assum_initial, as:eu:b:sig and as:eu:f:g hold. Then, there exists a unique strong solution to MV-SDE Mckean and the following estimate holds where $K>0$ is a constant that depends on $T$, $d$, $l$ and $L$.

Theorems & Definitions (48)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1: Existence and Uniqueness
  • proof
  • Theorem 3.1: Existence and uniqueness of IPS
  • proof
  • Theorem 3.2: Dimension-dependent PoC convergence rate
  • proof
  • Remark 3.1
  • Theorem 3.3: Sharp PoC rate across dimension
  • ...and 38 more