  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.
