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

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.