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On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with common noise

Ke Xu, Fen-Fen Yang, Chenggui Yuan

TL;DR

The paper addresses the well-posedness, long-time behavior, and numerical approximation of Lévy-driven McKean-Vlasov SDEs with common noise on an infinite horizon. It establishes existence, uniqueness, and moment bounds under non-globally Lipschitz coefficients via a contraction in the space of probability measures, and proves propagation of chaos for the associated particle system with explicit rates. A Tamed-Adaptive Euler-Maruyama (TAEM) scheme is proposed and analyzed, including discrete and continuous-time formulations, with rigorous proofs of finite-time reachability, moment bounds, and strong convergence on finite horizons. Numerical experiments across four models with jumps and common noise demonstrate near-first-order convergence and highlight the critical role of common noise in ensuring stability over long time scales, corroborating the theoretical findings and illustrating practical performance.

Abstract

In this work, we establish the existence and uniqueness of solutions to McKean-Vlasov stochastic differential equations (SDEs) driven by Lévy processes with common noise on an infinite time horizon, by means of a contraction mapping principle in the space of probability measures. In addition, we analyse the propagation of chaos for Lévy-driven McKean-Vlasov SDEs in the presence of common noise.

On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with common noise

TL;DR

The paper addresses the well-posedness, long-time behavior, and numerical approximation of Lévy-driven McKean-Vlasov SDEs with common noise on an infinite horizon. It establishes existence, uniqueness, and moment bounds under non-globally Lipschitz coefficients via a contraction in the space of probability measures, and proves propagation of chaos for the associated particle system with explicit rates. A Tamed-Adaptive Euler-Maruyama (TAEM) scheme is proposed and analyzed, including discrete and continuous-time formulations, with rigorous proofs of finite-time reachability, moment bounds, and strong convergence on finite horizons. Numerical experiments across four models with jumps and common noise demonstrate near-first-order convergence and highlight the critical role of common noise in ensuring stability over long time scales, corroborating the theoretical findings and illustrating practical performance.

Abstract

In this work, we establish the existence and uniqueness of solutions to McKean-Vlasov stochastic differential equations (SDEs) driven by Lévy processes with common noise on an infinite time horizon, by means of a contraction mapping principle in the space of probability measures. In addition, we analyse the propagation of chaos for Lévy-driven McKean-Vlasov SDEs in the presence of common noise.

Paper Structure

This paper contains 21 sections, 10 theorems, 247 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Well-posedness of Solutions
  3. Case $\gamma \neq 0$.
  4. Case $\gamma = 0$.
  5. Propagation of chaos
  6. Propagation of chaos
  7. Tamed-Adaptive Euler-Maruyama (TAEM) Method
  8. Discrete-time and Continuous-time TAEM Method
  9. Reachability of $\Delta_{t_k}$ over time $T$
  10. Step 1: Itô's formula for $|X^\Delta(t)|^l$.
  11. Step 2: Drift term estimation.
  12. Step 3: Diffusion terms estimation.
  13. Step 4: Jump term estimation.
  14. Step 5: Taking expectations and Grönwall's inequality.
  15. Convergence of TAEM Method under assumptions (A1)-(A7)
  16. ...and 6 more sections

Key Result

Theorem 2.3

Let assumptions (A1)-(A7) hold. Then, for any random distribution $X^\mu_0 \in \mathcal{F}_0$ with $\mathbb{E}|X^\mu_0|^2<\infty$, there exists a unique c$\grave{a}$dl$\grave{a}$g process X taking values in $\mathbb{R}^d$ satisfying the McKean-Vlasov SDE eq:MKV-SDE with initial distribution $X^\mu_0 where $K := K(\mathbb{E}|X^\mu_0|^2, d, L, L_1, \gamma_1, \gamma_2, T)$ is a positive constant.

Figures (1)

  • Figure 1: $\log_2\mathrm{MSE}(l,T)$ against $l$ for the four models ($T\!\in\!\{1,5,10\}$); dashed line: slope $-1$.

Theorems & Definitions (23)

  • Remark 2.1
  • Example 1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • ...and 13 more