Euler scheme for stochastic functional differential equations driven by fractional Brownian motion

TL;DR

This paper develops a rough paths-based Euler scheme to approximate solutions of stochastic functional differential equations driven by fractional Brownian motion with Hurst parameter greater than 1/2, achieving a specific convergence rate.

Contribution

It introduces a novel Euler approximation method for these equations using rough paths and Young integrals, with proven convergence rates and numerical validation.

Findings

Convergence rate of the scheme is $1/n^{ ext{gamma}}$ for any $ ext{gamma}<2 ext{lambda}-1$.

Numerical simulations confirm the theoretical convergence rates.

The method effectively handles fractional Brownian motion with $H>1/2$.

Abstract

In this paper, we apply rough paths techniques to provide an approximation of the solution of stochastic functional differential equations driven by fractional Brownian motion with Hurst parameter $H>1/2$. Here, the involved stochastic integral is the Young one and the coefficient is evaluated in the set of $\lambda$-H\"older continuous functions on $[-\tau,0]$, for some suitable $\tau>0$ and $\lambda\in(1/2,H)$. The rate of convergence of our scheme is $1/n^{\gamma}$, for any $\gamma<2\lambda-1$. Also, numerical simulations are provided to illustrate our theoretical results.