Malliavin calculus for fractional delay equations

TL;DR

This paper establishes the existence, estimates, and smooth density properties of solutions to fractional delay differential equations driven by H"older continuous functions, using Young integration and Malliavin calculus.

Contribution

It introduces a novel approach combining Young integration and Malliavin calculus to analyze fractional delay equations driven by fractional Brownian motion.

Findings

Existence and uniqueness of solutions for Young delay differential equations.

Solutions driven by fBm with H>1/2 have smooth probability densities.

Provides estimates for solutions of fractional delay equations.

Abstract

In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a H\"older continuous function with parameter greater that 1/2 via the Young integration setting. Then some estimates of the solution are obtained, which allow to show that the solution of a delay differential equation driven by a fractional Brownian motion (fBm) with Hurst parameter H>1/2 has a smooth density. To this purpose, we use Malliavin calculus based on the Frechet differentiability in the directions of the reproducing kernel Hilbert space associated with fBm.