### ON THE φ-ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS

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

*Theory of Stochastic Processes*

*Vol. 14 (30), no. 1, 2008, pp. 11–29*

In this paper we study the a.s. asymptotic behaviour of the solution of the stochastic differential equation

*dX(t) = g(X(t))dt +σ(X(t))dW(t), X(0) = b > 0,*

where

*g*and

*σ*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, σ*and

*φ,*under which

*φ(X(·))*can be approximated a.s. by

*φ(μ(·)),*where

*μ*is the solution of the ordinary differential equation

*dμ(t) = g(μ(t))dt, μ(0) = b*. As an application of these results we discuss the problem of

*φ-*asymptotic equivalence for solutions of stochastic differential equations.

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