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

TL;DR

This paper demonstrates that strong solutions to Itô's stochastic equations can be constructed on any probability space using approximation methods, correcting previous errors and introducing new convergence results for tamed Euler schemes.

Contribution

It revisits and corrects earlier proofs of strong solution existence, and introduces new convergence results for tamed Euler approximations in SDEs with unbounded drifts.

Findings

Strong solutions can be constructed on any probability space.

Corrected errors in previous proofs of key lemmas.

Established convergence of tamed Euler schemes for SDEs with unbounded drifts.

Abstract

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.