We present some recent results on optimal control/differential games in cases where the state equation is a stochastic delay differential equation (SDDE) with delay in the state and/or in the control. We look at two possible approaches: one based on partial smoothing properties of the transition semigroup associated with the uncontrolled SDDE (papers on optimal control , mean field control, mean field games without common noise with G. Bolli, F. Masiero, M. Ricciardi, M. Rosestolato), one based on regularization of viscosity solutions (paper with F. De Feo, A. Swiech, L. Wessels, on mean field control with only common noise).
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006