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Our subject of study is strong approximation of stochastic differential equations (SDEs) with respect to the supremum and the L_p error criteria, and we seek approximations that are strongly asymptotically optimal in specific classes of approximations. For the supremum error, we prove strong asymptotic optimality for specific tamed Euler schemes relating to certain adaptive and to equidistant time discretizations. For the L_p error, we prove strong asymptotic optimality for specific tamed Milstein schemes relating to certain adaptive and to equidistant time discretizations. To illustrate our findings, we numerically analyze the SDE associated with the Heston–3/2–model originating from mathematical finance.