Quantify the relation between countable- and continuous-innovation models

Quantify the relation between the optimal costs and policies of the countable-innovation surrogate model and those of the original continuous-innovation linear-Gaussian dual control problem.

Background

The paper replaces the continuous Gaussian innovation in the linear-Gaussian dual control problem with a countable-support innovation having the same first two moments. This surrogate preserves the posterior-moment identity needed for the weighted sup-norm contraction analysis and simplifies conditional expectations to countable sums, but it is not equivalent to the original continuous-innovation model.

The authors explicitly identify as unresolved the relationship between the optimal cost and policy obtained under the surrogate innovation model and those of the original continuous-Gaussian problem. Establishing quantitative approximation or comparison guarantees would clarify when the tractable countable-innovation analysis faithfully represents the original dual control problem.