Dual effect, certainty equivalence, and separation in stochastic control

Key Points

• To delineate the operational differences between feedback and closed-loop policies and establish the relationship between control dual effects, neutrality, and certainty equivalence in stochastic control.
• Analyzed fixed end-time stochastic control policies to contrast passively adaptive feedback mechanisms with actively adaptive closed-loop mechanisms.
• Mathematically evaluated the role of the dual effect, where control signals concurrently alter state trajectories and state uncertainty.
• Constructed theoretical proofs linking controller neutrality (the absence of a dual effect) to the certainty equivalence property.
• Demonstrated that closed-loop policies achieve active adaptation specifically when control inputs exhibit a dual effect on both the state and uncertainty.
• Proved an intimate mathematical connection between system neutrality and certainty equivalence.
• Extended the theoretical boundary and expanded the class of stochastic control problems known to satisfy certainty equivalence.