PRV property of functions and the asymptotic behaviour of solutions of stochastic differential equations

In this paper, we investigate the a.s. asymptotic behaviour of the solution of the stochastic differential equation $dX(t) = g(X(t))\,dt + \sigma(X(t))\,dW(t)$, where $g(\boldsymbol\cdot)$ and $\sigma(\boldsymbol\cdot)$ are positive continuous functions and $W(\boldsymbol\cdot)$ is a standard Wiener process. By an application of the theory of PRV and PMPV functions, we find conditions on $g(\boldsymbol\cdot)$ and $\sigma(\boldsymbol\cdot)$, under which $X(\boldsymbol\cdot)$ may be approximated a.s. on $\{X(t)\to\infty\}$ by the solution of the deterministic differential equation $d\mu(t) = g(\mu(t))\,dt$. Moreover, we study the asymptotic stability with respect to initial conditions of solutions of the above SDE as well as the asymptotic behaviour of generalized renewal processes connected with this SDE.