Existence of density for the solution of stochastic delay differential equations with reflection driven by a fractional Brownian motion

TL;DR

This paper proves the existence of a probability density for solutions of one-dimensional stochastic delay differential equations with reflection, driven by fractional Brownian motion with Hurst parameter greater than 1/2, using pathwise Riemann-Stieltjes integrals.

Contribution

It establishes the existence of a density for the law of solutions to reflected stochastic delay differential equations driven by fractional Brownian motion, a novel result in this context.

Findings

Density exists for the solution law.

Applicable to fractional Brownian motion with H > 1/2.

Uses pathwise Riemann-Stieltjes integrals.

Abstract

In this note we prove the existence of a density for the law of the solution for 1-dimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter $H > 1/2$. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.