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

In this paper we establish a stochastic maximum principle (SMP) for controlled stochastic differential equations with delay in the state and control-dependent noise, where the control set is not assumed to be convex. The cost functional includes a running and terminal costs depending on both the present state and its weighted past through general finite measures. Initially, we assume that the delay measures have square-integrable densities, allowing a reformulation of the problem as an infinite-dimensional stochastic evolution equation in a Hilbert space. Necessary conditions for optimality are derived using spike variation techniques and, under this framework, first- and second-order backward stochastic differential equations (BSDEs). The second-order adjoint process evolves in the space of Hilbert-Schmidt operators, yet only its finite-dimensional component is involved in the final SMP. When the delay is modeled by general finite measures (not necessarily absolutely continuous), the analysis avoids the infinite-dimensional reformulation. Instead, it relies on anticipated BSDEs to represent the first-order adjoint process, while approximating the measures to handle the second-order variation in a weak convergence framework. Here we assume that the terminal cost depends solely on the present state, so that we avoid to introduce a more general anticipated backward equations, which would raise the technical level. Furthermore, when the delay measures admit densities in Lp we show that the limiting second-order adjoint component satisfies a generalized mild backward equation, even though the underlying functional space lacks a Hilbertian structure. This provides additional insight into the structure of the SMP and suggests that the proposed methodology remains effective even outside the standard Hilbert space setting.