An Osgood's criterion for a semilinear stochastic differential equation

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 $\sigma \equiv 1$.