Existence of strong solutions for Itô's stochastic equations via approximations. Revisited

Abstract

Given strong uniqueness for an Itô'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.

21 pages