Stochastic maximum principle for problems with delay with dependence on the past through general measures

Abstract

<p style='text-indent:20px;'>We prove a stochastic maximum principle for a control problem where the state equation is delayed both in the state and in the control, and both the running and the final cost functionals may depend on the past trajectories. The adjoint equation turns out to be a new form of linear anticipated backward stochastic differential equations (ABSDEs in the following), and we prove a direct formula to solve these equations.</p>