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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