# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2020048

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

 1 Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9, 20133 Milano, Italia 2 Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, via Cozzi 55, 20125 Milano, Italia

* Corresponding author: Federica Masiero

Received  February 2020 Revised  September 2020 Published  November 2020

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.

Citation: Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020048
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