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

We consider the a.s. asymptotic behaviour of a solution of the stochastic differential equation (SDE) dX(t) = g(X(t))dt + σ(X(t))dW(t), with X(0) ≡ b > 0, where g(.) and σ(.) are positive continuous functions and W(.) is the standard Wiener process. By applying the theory of PRV and PMPV functions, we find the conditions on g(.) and σ(.), under which X(.) resp. f(X(.)) may be approximated a.s. on {X(t)→∞} by μ(.) resp. f(μ(.)), where μ( ) is a solution of the deterministic differential equation dμ(t) = g(μ(t))dt with μ(0) = b, and f(.) is a strictly increasing function. Moreover, we consider the asymptotic behaviour of generalized renewal processes connected with this SDE