For linear-convex extended mean field control, piecewise constant controls approximate the optimal cost at rate 1/2 and the optimal control at rate 1/4; under smoothness, the rate improves to 1.
Well-posedness of mean-field type forward-backward stochastic differential equations
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Convergence Rates of Time Discretization in Extended Mean Field Control
For linear-convex extended mean field control, piecewise constant controls approximate the optimal cost at rate 1/2 and the optimal control at rate 1/4; under smoothness, the rate improves to 1.