Lithuanian Mathematical Journal, Vol. 47, No. 4, 2007 PRV PROPERTY AND THE ϕ -ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS V. V. Buldygin Department of Mathematical Analysis and Probability Theory National Technical University of Ukraine, pr. Peremogy, 37, Kyiv 03056, Ukraine (e-mail: vbuldygi@mail.ru) O. I. Klesov Department of Mathematical Analysis and Probability Theory National Technical University of Ukraine, pr. Peremogy, 37, Kyiv 03056, Ukraine (e-mail: tbimc@ln.ua) J. G. Steinebach Universität zu Köln, Mathematisches Institut, Weyertal 86–90, D–50931 Köln, Germany (e-mail: jost@math.uni-koeln.de) Abstract. In this paper, we investigate the a.s. asymptotic behavior of the solution of the stochastic differential equation dX(t) = g(X(t)) dt + σ(X(t))dW(t), X(0) ≡ 1, where g(·) and σ(·) are positive continuous functions, and W(·) is a stan- dard Wiener process. By means of the theory of PRV functions we find conditions on g(·), σ(·), and ϕ(·) under which ϕ(X(·)) may be approximated a.s. by ϕ(µ(·)) on {X(t) → ∞}, where µ(·) is the solution of the ordinary differential equa- tion dµ(t) = g(µ(t)) dt with µ(0) = 1. Keywords: stochastic differential equation, asymptotic behavior of solutions, pseudo-regulary varying function. 1. INTRODUCTION Gikhman and Skorokhod [12], §17, Keller et al. [15], and later Buldygin et al. [8] considered the asymptotic behavior, as t →∞, of a solution X(·) = (X(t),t  0) of the stochastic differential equation (SDE) dX(t) = g(X(t)) dt + σ(X(t))dW(t), t  0, X(0) ≡ b> 0. (1.1) Here W(·) is a standard Wiener process, and X(·) denotes the Itô-solution of the SDE (1.1). One of the basic assumptions in the above works is that σ(·) = (σ(x), −∞ <x< ∞) is a positive function and that This work has partially been supported by Deutsche Forschungsgemeinschaft under DFG grants 436 UKR 113/41/0-3 and 436 UKR 113/68/0-1. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 445–465, October–December, 2007. Original article submitted September 21, 2007. 0363–1672/07/4704–0361 © 2007 Springer Science+Business Media, Inc. 361