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

Area of Science:

• Stochastic Optimal Control
• Mathematical Finance
• Control Theory

Background:

• Optimal control problems often involve complex systems with time delays.
• Terminal state constraints add significant challenges to these problems.
• Stochastic differential delayed equations model systems with inherent randomness and delays.

Purpose of the Study:

• To develop a method for solving stochastic optimal control problems with terminal state constraints.
• To introduce a time-delayed backward stochastic differential equation as an equivalent system.
• To derive a stochastic maximum principle for these constrained systems.

Main Methods:

• Formulating an equivalent backward delayed system.
• Utilizing Ekeland's variational principle.
• Applying the derived principle to specific models.

Main Results:

• A stochastic maximum principle is successfully obtained for the addressed problem class.
• The method is demonstrated through applications to linear-quadratic control and economic models.
• The study provides a theoretical framework for state-constrained delayed stochastic control.

Conclusions:

• The derived stochastic maximum principle offers a powerful tool for analyzing state-constrained delayed stochastic control problems.
• The approach is versatile, applicable to both theoretical control models and practical economic scenarios.
• This research advances the understanding and solution techniques for complex stochastic control systems.