On the positivity of the density of stochastic delay differential equations driven by a fractional Brownian motion

TL;DR

This paper proves the strict positivity of the density for solutions to stochastic delay differential equations driven by fractional Brownian motion, under certain smoothness and ellipticity conditions, using advanced Gaussian bounds.

Contribution

It introduces a novel combination of methods to establish the positivity of the density for such equations, extending previous results to fractional Brownian motion settings.

Findings

Density of the solution is strictly positive in its support.

Provides Gaussian-type lower bounds for the density.

Extends positivity results to fractional Brownian motion-driven equations.

Abstract

In this paper, we consider a Stochastic Delay Differential Equation with constant delay $r>0$ and, under the same conditions on the coefficients needed to ensure the smoothness of the density plus an ellipticity condition on the diffusion term, we prove that the density function of the solution is strictly positive in its support. In order to prove it, we give a Gaussian-type lower bound for the density of the solution combining the Nourdin and Viens' density bounding method together with Kohatsu-Higa's method.