Malliavin Calculus for Degenerate Stochastic Functional Differential Equations

Abstract

Consider the solution {X(t); t∈[−r,T]} of the following stochastic functional differential equation:

$$dX(t)=\biggl\{\int_{-r}^{0}\rho(s)X(t+s)\,ds+A_{0}(t,X(t))\biggr\}dt+\sum_{i=1}^{m}A_{i}(t,X(t))\,dW^{i}(t),$$

where ρ(t) is an ℝ-valued function on [−r,0], and {W(t); t∈[0,T]} is an m-dimensional Brownian motion. The main purpose is to study the smoothness of the probability density of X(T) with respect to the Lebesgue measure.