Existence of strong solutions for It\^o's stochastic equations via approximations. Revisited

Given strong uniqueness for an It\^o's stochastic equation, we prove that its solution can beconstructed on "any" probability space by using, for example, Euler's polygonal approximations. Stochastic equations in $\mathbb{R}^{d}$ and in domains in $\mathbb{R}^{d}$ are considered. This is almost a copy of an old article in which we correct errors in the original proof of Lemma 4.1 found by Martin Dieckmann in 2013. We present also a new result on the convergence of "tamed Euler approximations" for SDEs with locally unbounded drifts, which we achieve by proving an estimate for appropriate exponential moments.