Convergence of delay differential equations driven by fractional Brownian motion

TL;DR

This paper establishes existence, uniqueness, and convergence results for delay differential equations driven by fractional Brownian motion with Hurst parameter greater than 1/2, using pathwise integrals.

Contribution

It provides the first rigorous proof of solution convergence for stochastic delay differential equations driven by fractional Brownian motion as delay vanishes.

Findings

Solutions exist and are unique for the considered equations.

Solutions converge to the non-delay case as delay tends to zero.

The convergence holds almost surely and in L^p.

Abstract

In this note we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter $H > 1/2$. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in $L^p$, to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.