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Strong convergence of tamed theta scheme for superlinearly growing McKean-Vlasov NSDDEs driven by fractional Brownian motions

Li Tan, Shizhong Hu, Shengrong Wang

TL;DR

This work advances numerical analysis for McKean–Vlasov neutral stochastic delay equations driven by fractional Brownian motion with super-linear drift by introducing a tamed theta Euler–Maruyama scheme. It proves well-posedness (existence and uniqueness) via Picard iterations, establishes propagation of chaos for the associated interacting particle system, and derives rigorous strong convergence rates for the scheme. The analysis leverages fractional Itô calculus, fractional BDG inequalities, and Mittag–Leffler-type bounds to handle memory and non-Lipschitz drift. The results provide a robust, provably convergent numerical method for simulating fractional MV NSDDEs with delays and memory effects, with explicit rates depending on the taming parameter $\alpha$ and the Hurst index $H$.

Abstract

In this article, we study the McKean-Vlasov neutral stochastic differential delay equations driven by fractional Brownian motion with super-linearly growing coefficients, where the Hurst exponent $H\in(1/2,1)$. The existence and uniqueness of the exact solution were shown by the Picard iteration. Besides, we propose a tamed theta Euler-Maruyama scheme for this equation, analyzed the moment boundness and propagation of chaos etc. Moreover, the convergence rate of the numerical scheme is established.

Strong convergence of tamed theta scheme for superlinearly growing McKean-Vlasov NSDDEs driven by fractional Brownian motions

TL;DR

This work advances numerical analysis for McKean–Vlasov neutral stochastic delay equations driven by fractional Brownian motion with super-linear drift by introducing a tamed theta Euler–Maruyama scheme. It proves well-posedness (existence and uniqueness) via Picard iterations, establishes propagation of chaos for the associated interacting particle system, and derives rigorous strong convergence rates for the scheme. The analysis leverages fractional Itô calculus, fractional BDG inequalities, and Mittag–Leffler-type bounds to handle memory and non-Lipschitz drift. The results provide a robust, provably convergent numerical method for simulating fractional MV NSDDEs with delays and memory effects, with explicit rates depending on the taming parameter and the Hurst index .

Abstract

In this article, we study the McKean-Vlasov neutral stochastic differential delay equations driven by fractional Brownian motion with super-linearly growing coefficients, where the Hurst exponent . The existence and uniqueness of the exact solution were shown by the Picard iteration. Besides, we propose a tamed theta Euler-Maruyama scheme for this equation, analyzed the moment boundness and propagation of chaos etc. Moreover, the convergence rate of the numerical scheme is established.

Paper Structure

This paper contains 7 sections, 13 theorems, 171 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Existence and Uniqueness
  4. Propagation of Chaos
  5. The Tamed Theta EM Scheme
  6. Moment Boundness
  7. Strong Convergence Rate

Key Result

Lemma 2.1

(Fractional Itô formula) Let $X_{t}$ be the solution of fbmitofunction with $1/2<H<1$, and assume that $\mathbb{E}[\int_{0}^T\left\lvert F(s,X_{s})D_{s}^{\phi}\int_{0}^sF(u,X_{u}){\rm d}B_{u}^H\right\rvert{\rm d}s]<\infty$, $\frac{\partial V(s,X_s)}{\partial X}F(s,X_s)\in L_{\phi}(0,T)$ for $V\in C^ where $F^T$ is the transpose of $F$, and $\text{\rm tr}(\cdot)$ denotes the matrix trace.

Theorems & Definitions (28)

  • Definition 2.1
  • Lemma 2.1
  • Remark 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Remark 2.9
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • ...and 18 more