  We consider an Ito stochastic differential equation with delay, driven by
brownian motion, whose solution, by an appropriate reformulation, defines a
Markov process $X$ with values in a space of continuous functions $\mathbf C$,
with generator $\mathcal L$. We then consider a backward stochastic
differential equation depending on $X$, with unknown processes $(Y,Z)$, and we
study properties of the resulting system, in particular we identify the process
$Z$ as a deterministic functional of $X$. We next prove that the
forward-backward system provides a suitable solution to a class of parabolic
partial differential equations on the space $\mathbf C$ driven by $\mathcal L$,
and we apply this result to prove a characterization of the fair price and the
hedging strategy for a financial market with memory effects. We also include
applications to optimal stochastic control of differential equation with delay:
in particular we characterize optimal controls as feedback laws in terms the
process $X$.
