On the exact order of growth of solutions of stochastic differential equations with time-dependent coefficients

We study the exact order of growth of the solution of the stochastic differential equation $d\eta (t)=g \left(\eta (t)\right)\varphi (t)dt +\sigma \left(\eta (t)\right)\theta (t)dw(t),$ $X(0)=b,$ where $w$ is the standard Wiener process, $b$ is a nonrandom positive constant, $g$, $\sigma$ are continuous positive functions, and $\varphi$ and $\theta$ are real continuous functions such that a continuous solution $\eta$ exists. As an application of these results, we discuss the problem of asymptotic equivalence for solutions of stochastic differential equations.