  The purpose of this paper is to give an Osgood's criterion for solutions of
semilinear stochastic differential equations of the form $X_{t}=\xi
+\int_{0}^{t}b(s,X_{s})ds+\int_{0}^{t}\sigma (s)X_{s}dW_{s},\ t\geq 0$. Here,
$b$ is a non-negative, non-decreasing by components and continuous random field
and $\sigma $ is a predictable and continuous process. Also we present a
generalization of the so-called Feller's test whenever $\sigma \equiv 1$.
