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

TL;DR

This paper establishes a stochastic maximum principle for control problems with delays in state and control, where the cost depends on past trajectories, introducing a new form of anticipated backward stochastic differential equations and a direct solution method.

Contribution

It introduces a novel stochastic maximum principle for delayed control problems with general past dependence and provides a direct solution formula for the associated anticipated backward stochastic differential equations.

Findings

Derived a new stochastic maximum principle for delayed control problems

Formulated a novel class of linear anticipated backward stochastic differential equations

Provided a direct solution method for these ABSDEs

Abstract

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 also the final cost functional may depend on the past trajectories. The adjoint equations turn 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.