Stochastic maximum principle for SPDEs with delay

TL;DR

This paper establishes necessary optimality conditions, specifically a Pontryagin maximum principle, for controlling infinite-dimensional stochastic systems with delays, incorporating history-dependent costs via novel anticipated backward stochastic differential equations.

Contribution

It introduces a new form of anticipated backward stochastic differential equations to handle history-dependent costs in stochastic control of delay systems, extending the maximum principle to infinite dimensions.

Findings

Derived necessary conditions for optimal control with delays

Introduced a new class of anticipated backward stochastic differential equations

Extended maximum principle to infinite-dimensional stochastic systems with delay

Abstract

In this paper we develop necessary conditions for optimality, in the form of the Pontryagin maximum principle, for the optimal control problem of a class of infinite dimensional evolution equations with delay in the state. In the cost functional we allow the final cost to depend on the history of the state. To treat such kind of cost functionals we introduce a new form of anticipated backward stochastic differential equations which plays the role of dual equation associated to the control problem.