In this paper, we prove the existence and uniqueness of maximally defined strong solutions to SDEs driven by multiplicative noise on general space-time domains $Q\subset\mathbb{R}_+\times\mathbb{R}^d$, which have continuous paths on the one-point compactification $Q\cup\partial$ of $Q$ where $\partial \notin Q$ and $Q\cup\partial$ is equipped with the Alexandrov topology. If the SDE is of gradient type (see (2.5) below) we prove that under suitable Lyapunov type conditions the life time of the solution is infinite and its distribution has sub-Gaussian tails. This generalizes earlier work \cite{KR} by Krylov and one of the authors to the case where the noise is multiplicative.