Numerical scheme for delay-type stochastic McKean-Vlasov equations driven by fractional Brownian motion

TL;DR

This paper develops a numerical scheme for delay-type stochastic McKean-Vlasov equations driven by fractional Brownian motion, proving solution existence, propagation of chaos, and convergence of the Euler-Maruyama method.

Contribution

It introduces a novel numerical approach for DSMVEs with delay and fractional noise, establishing convergence rates and demonstrating practical simulation.

Findings

Existence and uniqueness of solutions proved.

Propagation of chaos established in $\\mathcal{L}^p$ sense.

Euler-Maruyama scheme effectively approximates systems with delay.

Abstract

This paper focuses on the numerical scheme for delay-type stochastic McKean-Vlasov equations (DSMVEs) driven by fractional Brownian motion with Hurst parameter $H\in (0,1/2)\cup (1/2,1)$. The existence and uniqueness of the solutions to such DSMVEs whose drift coefficients contain polynomial delay terms are proved by exploting the Banach fixed point theorem. Then the propagation of chaos between interacting particle system and non-interacting system in $\mathcal{L}^p$ sense is shown. We find that even if the delay term satisfies the polynomial growth condition, the unmodified classical Euler-Maruyama scheme still can approximate the corresponding interacting particle system without the particle corruption. The convergence rates are revealed for $H\in (0,1/2)\cup (1/2,1)$. Finally, as an example that closely fits the original equation, a stochastic opinion dynamics model with both extrinsic memory and intrinsic memory is simulated to illustrate the plausibility of the theoretical result.