  We consider a Markov chain approximation scheme for utility maximization
problems in continuous time, which uses, in turn, a piecewise constant policy
approximation, Euler-Maruyama time stepping, and a Gauss-Hermite approximation
of the Gaussian increments. The error estimates previously derived in Picarelli
and Reisinger (2019) are asymmetric between lower and upper bounds due to the
control approximation and improve on known results in the literature in the
lower case only. In the present paper, we use duality results to obtain a
posteriori upper error bounds which are empirically of the same order as the
lower bounds. The theoretical results are confirmed by our numerical tests.
