Malliavin differentiability and regularity of densities in semi-linear stochastic delay equations driven by weighted fractional Brownian motion

TL;DR

This paper establishes the existence, uniqueness, and smoothness of solution densities for semi-linear stochastic delay equations driven by weighted fractional Brownian motion, advancing understanding of their regularity properties.

Contribution

It demonstrates the existence, uniqueness, and density smoothness of solutions to semi-linear stochastic delay equations driven by weighted fractional Brownian motion.

Findings

Proved existence and uniqueness of solutions

Established smoothness of solution densities

Extended results to multi-dimensional cases (d ≥ 1)

Abstract

In this work, we will show the existence and uniqueness of the solution to the semi linear stochastic differential equations driven by weighted fractional Brownian motion with delay. We also prove smoothness of the density of the solution with respect to Lebesgue's measure on R^d for $d \geq 1$.