On the $\varphi$-asymptotic behaviour of solutions of stochastic differential equations

In this paper we study the a.s. asymptotic behaviour of the solution of the stochastic differential equation $dX(t)=g(X(t))dt+\sigma(X(t))dW(t), X(0)=b>0,$ where $g$ and $\sigma$ are positive continuous functions and $W$ is a Wiener process. Making use of the theory of pseudo-regularly varying (PRV) functions, we find conditions on $g, \sigma$ and $\varphi,$ under which $\varphi(X(\cdot))$ can be approximated a.s. by $\varphi(\mu(\cdot)),$ where $\mu$ is the solution of the ordinary differential equation $d\mu(t)=g(\mu(t))dt, \mu(0)=b.$ As an application of these results we discuss the problem of $\varphi$-asymptotic equivalence for solutions of stochastic differential equations.