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McKean-Vlasov stochastic differential equations with super-linear measure arguments: well-posedness and propagation of chaos

Zhuoqi Liu, Qian Guo, Shuaibin Gao, Chenggui Yuan

TL;DR

This paper addresses the strong well-posedness and mean-field (propagation of chaos) properties of McKean–Vlasov SDEs with super-linear growth in state and measure arguments. It develops an Euler-like approximation under a locally monotone condition to establish existence and uniqueness of a strong solution $X_t$, and analyzes finite- and infinite-horizon propagation of chaos for the interacting particle system converging to its non-interacting mean-field limit as $N\to\infty$ (and in time for the infinite horizon). The results are demonstrated on examples with high-degree drift terms and distributional dependencies, complemented by numerical simulations that corroborate the theoretical predictions. The framework relaxes global Lipschitz assumptions, using local Lipschitz in state and law with dissipativity to cover a broad class of distribution-dependent coefficients.

Abstract

This paper studies McKean-Vlasov stochastic differential equations (MVSDEs) whose drift coefficients grow super-linearly in both state variables and measure arguments, and whose diffusion coefficients exhibit super-linear growth in the state variables. By constructing an Euler-like sequence, we establish the strong well-posedness of such MVSDEs under a locally monotone condition. Furthermore, the propagation of chaos is studied on both finite and infinite horizons, demonstrating convergence of the interacting particle system to the corresponding non-interacting system. To illustrate the rationality of the theoretical results, we provide examples whose drifts contain the high powers and multiple integrals of distributions, with numerical simulations presented in Section 6.

McKean-Vlasov stochastic differential equations with super-linear measure arguments: well-posedness and propagation of chaos

TL;DR

This paper addresses the strong well-posedness and mean-field (propagation of chaos) properties of McKean–Vlasov SDEs with super-linear growth in state and measure arguments. It develops an Euler-like approximation under a locally monotone condition to establish existence and uniqueness of a strong solution , and analyzes finite- and infinite-horizon propagation of chaos for the interacting particle system converging to its non-interacting mean-field limit as (and in time for the infinite horizon). The results are demonstrated on examples with high-degree drift terms and distributional dependencies, complemented by numerical simulations that corroborate the theoretical predictions. The framework relaxes global Lipschitz assumptions, using local Lipschitz in state and law with dissipativity to cover a broad class of distribution-dependent coefficients.

Abstract

This paper studies McKean-Vlasov stochastic differential equations (MVSDEs) whose drift coefficients grow super-linearly in both state variables and measure arguments, and whose diffusion coefficients exhibit super-linear growth in the state variables. By constructing an Euler-like sequence, we establish the strong well-posedness of such MVSDEs under a locally monotone condition. Furthermore, the propagation of chaos is studied on both finite and infinite horizons, demonstrating convergence of the interacting particle system to the corresponding non-interacting system. To illustrate the rationality of the theoretical results, we provide examples whose drifts contain the high powers and multiple integrals of distributions, with numerical simulations presented in Section 6.
Paper Structure (12 sections, 7 theorems, 120 equations, 1 figure)

This paper contains 12 sections, 7 theorems, 120 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['thm1']}
  4. Proof of Theorem \ref{['keythm']}
  5. Proof of Theorem \ref{['longpoc']}
  6. Numerical simulation
  7. Appendix
  8. Proof of the exponential integrability stated in Remark \ref{['rree01']}
  9. Verification of coefficient conditions for Example (\ref{['exm']}) in Remark \ref{['numrem']}
  10. Verification of Assumption \ref{['ass1']}
  11. Verification of Assumption \ref{['ass3']}
  12. Verification of Assumption \ref{['ass6']}

Key Result

Theorem 2.6

Assume that Assumptions ass1-ass6 hold with $p \geq \max\{\gamma, 2 + 2l_3, 4l_4\}$. Then, for any $T>0$, the MVSDE (MV) admits a unique strong solution $X_t$. Moreover, for $2\leq\bar{p}\leq p/l_4$, it holds that

Figures (1)

  • Figure 1: Strong convergence with respect to the number of particles

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.10
  • proof
  • Lemma 3.1
  • proof
  • ...and 7 more