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

We consider an It̂o 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 C, with generator 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 C driven by 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 of the process X.