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In N.V. Krylov, Approximating value functions for controlled degeneratediffusion processes by using piece-wise constant policies, Electron. J.Probab., 4(2), 1999, it is proved under standard assumptions that the valuefunctions of controlled diffusion processes can be approximated with order 1/6error by those with controls which are constant on uniform time intervals. Inthis note we refine the proof and show that the provable rate can be improvedto 1/4, which is optimal in our setting. Moreover, we demonstrate theimprovements this implies for error estimates derived by similar techniques forapproximation schemes, bringing these in line with the best available resultsfrom the PDE literature.

Improved order 1/4 convergence for piecewise constant policy approximation of stochastic control problems