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We consider an Ito stochastic differential equation with delay, driven bybrownian motion, whose solution, by an appropriate reformulation, defines aMarkov process $X$ with values in a space of continuous functions $\mathbf C$,with generator $\mathcal L$. We then consider a backward stochasticdifferential equation depending on $X$, with unknown processes $(Y,Z)$, and westudy properties of the resulting system, in particular we identify the process$Z$ as a deterministic functional of $X$. We next prove that theforward-backward system provides a suitable solution to a class of parabolicpartial 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 thehedging strategy for a financial market with memory effects. We also includeapplications to optimal stochastic control of differential equation with delay:in particular we characterize optimal controls as feedback laws in terms theprocess $X$.

Stochastic equations with delay: optimal control via BSDEs and regular solutions of Hamilton-Jacobi-Bellman equations