Peng's maximum principle gives necessary optimality conditions for stochastic control problems with non-convex control sets and control-dependent noise, at the price of a second-order adjoint equation. In this talk I present two recent results, obtained jointly with G. Guatteri and L. Wessels, that extend it to systems with memory. The first concerns stochastic delay equations with general finite delay measures, including pointwise delay: the problem is lifted to a Hilbert space and, since the second-order adjoint is not well defined for general measures it is obtained as a limit of approximations. The second concerns delay equations of mean-field type, where the coefficients depend on the state, its past and their expectations: a sharp asymptotic estimate for the mean of the first-order variation allows one to neglect second-order derivatives in the law and yields Peng's principle.

The talk is based on joint works with G. Guatteri and L. Wessels