# Maximum principle for a stochastic delayed system involving terminal state constraints

## TL;DR

This paper develops a stochastic maximum principle for control systems with delays and terminal state constraints, using backward delayed equations and variational methods, with applications to linear-quadratic and economic models.

## Contribution

It introduces a novel maximum principle for delayed stochastic systems with terminal constraints, employing a backward delayed system and Ekeland's variational principle.

## Key findings

- Derived a stochastic maximum principle for delayed systems with terminal constraints
- Applied the principle to linear-quadratic control and economic production models
- Validated the theoretical results through illustrative examples

## Abstract

We investigate a stochastic optimal control problem where the controlled system is depicted as a stochastic differential delayed equation; however, at the terminal time, the state is constrained in a convex set. We firstly introduce an equivalent backward delayed system depicted as a time-delayed backward stochastic differential equation. Then a stochastic maximum principle is obtained by virtue of Ekeland's variational principle. Finally, applications to a state constrained stochastic delayed linear-quadratic control model and a production-consumption choice problem are studied to illustrate the main obtained result.