Stochastic maximum principle for equations with delay: going to infinite dimensions to solve the non-convex case

In this paper we develop necessary conditions for optimality, in the form of the stochastic Pontryagin maximum principle, for controlled equation with delay in the state and with control dependent noise, in the general case of controls $u \in U$ with $U$ not necessarily convex. The maximum principle is formulated by means of a first and second order adjoint BSDEs.