Convergence rate of nonlinear delayed neutral McKean-Vlasov SDEs driven by fractional Brownian motions

TL;DR

The paper addresses strong convergence for a neutral McKean–Vlasov SDDE driven by fractional Brownian motion with $H\in(1/2,1)$. It establishes well-posedness via Carathéodory approximation, analyzes propagation of chaos through a particle-system mean-field limit, and derives a strong convergence rate for Euler–Maruyama discretizations of the interacting system, with rate $\Delta^{H}$ in $L^p$ for $p>1/H$. A numerical example with $H=0.8$ confirms the theoretical rate. Overall, the work extends numerical analysis for MV-SDEs to neutral, delayed, nonlinear, and fractional-noise settings, offering practical insights for simulations involving long-range dependence.

Abstract

In this paper, our main aim is to investigate the strong convergence for a neutral McKean-Vlasov stochastic differential equation with super-linear delay driven by fractional Brownian motion with Hurst exponent $H\in(1/2, 1)$. After giving uniqueness and existence for the exact solution, we analyze the properties including boundedness of moment and propagation of chaos. Besides, we give the Euler-Maruyama (EM) scheme and show that the numerical solution converges strongly to the exact solution. Furthermore, a corresponding numerical example is given to illustrate the theory.

Figures (1)

• Figure 1: Mean square error of the EM method for equation \ref{['anli']} with $H=0.8$

Theorems & Definitions (24)

• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Lemma 4.1