Stochastic Maximum Principle for Equations with Delay: Going to Infinite Dimensions to Solve the Nonconvex 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 ofthe 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 relieson 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 introducing a more general anticipated backward equations, which would raise the technical level. Furthermore, when the delay measures admit densities in Lp([d, 0]), p ϵ (1, 2), 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.