The Malliavin calculus and stochastic delay equations

Abstract

We consider stochastic delay systems dx( t) = g( x( t − r)) dW( t) driven by multi-dimensional Brownian motion W. The diffusion coefficient g is smooth with a possible degeneracy at 0. For a large class of deterministic initial paths we show that the solution x( t) admits a smooth density with respect to Lebesgue measure. The proof is based on Malliavin calculus together with new probabilistic lower bounds on the solution x.