The Malliavin calculus and stochastic delay equations

AbstractWe 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