A sufficient maximum principle for backward stochastic systems with mixed delays

Area of Science:

• Stochastic Analysis
• Optimal Control Theory
• Numerical Simulation

Background:

• Backward stochastic differential equations (BSDEs) are crucial in finance and control.
• Incorporating time delays (discrete, moving-average, noisy memory) complicates BSDE analysis and control.
• Existing methods often struggle with multiple, complex delay types in BSDEs.

Purpose of the Study:

• To establish sufficient optimality conditions for stochastic systems with three types of delays.
• To introduce and validate novel time-advanced adjoint equations for analyzing these delayed BSDEs.
• To apply the developed theory to a linear-quadratic backward stochastic system and analyze delay impacts.

Main Methods:

• Derivation of sufficient optimality conditions for delayed BSDEs.
• Introduction of two equivalent time-advanced stochastic differential equations (adjoint equations).
• Malliavin calculus for handling derivatives in the adjoint equations.
• Discretization techniques for simulating delayed stochastic differential equations.

Main Results:

• Establishment of a sufficient optimality condition for the studied stochastic system.
• Demonstration of the equivalence between two types of time-advanced adjoint equations.
• Explicit optimal control derived for a linear-quadratic backward stochastic system.
• Numerical simulations illustrating the influence of time delays on control outcomes.

Conclusions:

• The proposed framework effectively addresses optimal control for BSDEs with multiple delays.
• The developed adjoint equations provide a viable tool for analyzing complex delayed stochastic systems.
• Time delays significantly impact the optimal control strategies and solutions in these systems.