### PRV PROPERTY AND THE ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS

**V. V. BULDYGIN, O. I. KLESOV, AND J. G. STEINEBACH**

*Theory of Stochastic Processes Vol. 11 (27), no. 3–4, 2005, pp. 42–57*

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.

*φ(X(·))*may be approximated a.s. on

*{X(t)→∞}*by

*μ(·)*resp.

*φ(μ(·))*, where

*μ(·)*is a solution of the deterministic differential equation

*dμ(t) = g(v(t))dt with μ(0) = b*, and

*φ(·)*is a strictly increasing function. Moreover, we consider the asymptotic behaviour of generalized renewal processes connected with this SDE.

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