Stochastic Maximum Principle for Optimal Control of Anticipated Backward Stochastic Systems with Delays

TL;DR

This paper develops a generalized stochastic maximum principle for optimal control of delayed backward stochastic systems with anticipation, extending previous bounded delay models to include long-term memory and forward-looking effects.

Contribution

It introduces a new formulation using $\sigma$-finite measures for delayed systems and derives necessary and sufficient conditions for optimal control in this framework.

Findings

Derived explicit optimal controls for climate policy, consumption, and linear quadratic problems.

Extended maximum principle to systems with unbounded delays and anticipation.

Validated the theoretical results with illustrative examples.

Abstract

This paper investigates optimal control problems for delayed systems governed by Infinitely Anticipated Backward Stochastic Differential Equations (IABSDEs). Unlike existing frameworks limited to bounded delays, we introduce a generalized formulation utilizing $\sigma$-finite measures that accommodates both long-term memory effects and forward-looking anticipation. Employing a new type of infinitely delayed stochastic differential equations as adjoint equations, we derive the necessary conditions of the maximum principle for optimal control. Under appropriate assumptions, the sufficiency of the maximum principle is also established. As illustrative examples, a climate policy model, a consumption optimization problem and a linear quadratic control problem are discussed, and all optimal controls are derived explicitly.